Kheldians are an alien species that was introduced to the game as part of the backstory for heroic "Epic" Archetypes. The playable Kheldians are Peacebringers and Warshades; the villainous Epic Archetypes are Soldiers and Widows. These archetypes are intended to have their own storyline, or "epic," to follow from levels 1-50; to that end, they have special contacts and missions that are unlocked as players level up. Because of the archetype-specific storyline, players that wish to roleplay a Kheldian accurately according to the game lore have less freedom with a character's backstory than other archetypes. Kheldians are also designed to be a "jack of all trades" style of archetype. Because of this, they are locked in to two powersets that have many more abilities than traditional powersets, and give players different shapeshifting forms to choose from. The changes to their powersets and especially the different forms give the Kheldians a significantly different play style from other archetypes.
Kheldians are energy beings from outside our galaxy. They have the ability to join with human hosts in a (usually) symbiotic relationship. When this happens, the two separate personalities merge into one new one with its own identity and name. There are three types you'll typically encounter in the City of Heroes universe: Peacebringers, Nictus and Warshades.
Peacebringers are regular Kheldians, beings composed of light. The Peacebringers themselves are dedicated to hunting down the Nictus and destroying them for what they did to the Kheldian race (and still plan on doing.)
Nictus are Kheldians who, when faced with the short native lifespan of the Kheldian people, turned to science and changed themselves into beings of darkness. They feed off of life energy, and were at one point able to siphon the life force of the Kheldians from great distances.
"I have told you that the Nictus once fed upon other Kheldians. What I did not tell you was how. The Nictus scientists developed a powerful energy transfer device that could rip away a Kheldian's life force, even at a distance of light years. This terrible weapon is the reason we Peacebringers have resolved to destroy all Nictus, wherever they may be." - Moonfire
Warshades are "redeemed" Nictus. These are Nictus that have had a change of heart, merging willingly with another instead of trying to force their way in. They fight alongside the Peacebringers against the Nictus, and have a sometimes uneasy truce with them.
The native lifespan of an unjoined Kheldian is 10 years. Some have said the Nictus lifespan was reduced to 5 years, but this is unverified.
Kheldians can, however, merge with a host being. They aren't (typically) parasitic, though the Nictus often seem to try to take over the host's mind. With a continual chain of hosts, Kheldians are functionally immortal. (Shadowstar, the Warshade leader, has been around since ancient Egypt - 3000 to 5000 years.)
Nictus, after traveling across vast distances via Shadow Cysts, tend to be weak and use the cysts as "life support" if no host is ready. If the cyst is destroyed, the Nictus die off quickly, regardless of age.
Most beings are, for lack of a better description, like apartments to Kheldians. They can move in, if the host is willing (or very weak,) and they can separate at will. They cannot be forced into someone (again, unless they're very weak, or "boosted" in as Arakhn tried to do.) And with enough willpower, the host can force them out. No 30 day notice needed.
The joining has advantages for both the host and the Kheldian.
The host gets power - typically seen in-game as attacks, shields and the like. In addition, it seems the host's natural lifespan is increased as well.
The Kheldian, as mentioned, stretches its lifespan when joined. In addition, in the future it can make an energy "copy" of any being it's been joined with in the past - such as the Nova and Dwarf forms of tri-form Kheldians. (So, yes, in the future a Kheldian from Paragon could merge with something else and change to Human form. Or alien, or catgirl, or whatever else it merged with.)
The two personalities merge together into a new single entity. In the case of Peacebringers and Warshades the personalities influence the whole. Nictus, on the other hand, do their best to dominate the host and take complete control. It is possible for the Kheldian to depart the host and the host can actually force a Kheldian out with enough focused willpower.
Yes, forms. Being energy, Kheldians have no natural "shape." At best they appear as wisps of energy. OK, occasionally bad tempered, deadly wisps of energy, but wisps nonetheless. You can see this if you run across a Shadow Cyst - it'll spout "puffs" called Unbound Nictus that will do their best to kill you. They're pretty good at it when they get going.
Most people that have seen a Kheldian are used to two non-host forms:
The Nova form, also known as the "squid," is a flying, tentacled form (no relation to the FSM) based on a life form the Kheldians encountered living in a gas giant. It has inherent flying ability - not much to stand on in a gas giant, after all.
There are two "types" of Nova, the Bright and Dark Novas. They're essentially the same, and best thought of much like Human races - just because you're Asian, Caucasian, African or whatnot, you're still human. The Bright Novas are from the gas giant's upper reaches, and are called Mefnanim. The Dark Novas are from the depths of the same planet, and called Hulmanim.
The Dwarf form, also called the "lobster," is a slow moving, tough form that is derived from creatures that lived on the surface of a very dense dwarf star. It's very durable, and has an inherent self-teleport ability.
Once more, two "types" or "races." The White Dwarf form's "base" race are called the Kurukt, and lived on neutron stars. The Black Dwarf form is based on the Ruktur, who are said to have lived on the surface of a pulsar. Yes, we know how small those are. It's a comic-book world.
Also, as mentioned, it appears the Kheld can make a copy of anything it's joined with previously. Judging from story arcs and in-game canon, it's an individual memory, not a racial memory. This also lets "all Human" or bi-form (Human/Dwarf, Human/Nova) Khelds explain why they don't have "all" their forms. We don't know if there's a limit to the number of forms they can assume or not - it's possible a Kheldian could visit, for example, three other planets, have five more species of hosts, and become octoform.
Other Effects on the Host
Well, other than glowing eyes, it's hard to say. Given this is in City of Heroes, people with odd colored skin (or made up color) aren't exactly rare. Given there's no consistent coloration, it's a safe bet it's part of a costume.
In addition, when forcibly separated at least, the two halves can "feel" the other's out there. We don't know if this happens with a willing separation or not.
"They used their magic, and they pulled us apart. They pulled Altered Umbra out of Lars Medelson, and trapped it in a crystal. The Kheldian is gone, and so is part of me. I can still feel Altered Umbra, my Kheldian half, out there somewhere. The rituals they're doing here will sever that connection forever. You've got to stop them! And you've got to rescue Altered Umbra! I know where he is. I wish I could help you, but I'm... I'm not Shadowcatcher anymore. I'm not sure I'm even Lars Mendelson." - Lars Mendelson
We don't know if this is from the magic, a side effect of forcible separation, or if it "always" happens.
For the Peacebringers, the top one We know of is Sunstorm. He seems to have some clout, as Shadowstar refers to him when mentioning the truce between the Peacebringers and Warshades. Moonfire, however, seems to be the Kheld "PR" department and liaison with other superheroes and hero groups.
As for a "ruling body," the Kheldians ARE extragalactic. While there may be a "ruling council" or something similar, the Peacebringers on earth don't seem to refer to them.
"When we are done and this world is under our rule, the Nictus will need a leader if we are to stave off the certain Peacebringer assault and quell the growing number of Warshades. If you show such weakness again, you shall not live to challenge me for the title."
so it's hardly a "secure" position. Arakhn's also in charge of the Galaxy troops and Void Hunter mercenaries, while Requiem has the War Wolves.
Shadowstar is not the leader but a helper similar to Sunstorm.
How long have they been here?
A very long time. As mentioned before, Shadowstar has been around since the time of ancient Egypt. The Nictus have definitely been here at least as long, given references to the Path of the Dark (using Shadow Seeds - essentially Nictus Cysts) being here since Ancient Rome - and they sounded pretty well established by that time. (The cyst in Ravenna mentioned in the PotD arc has been there since the 5th century AD.)
How long have the Peacebringers been here? We don't know. There are many scattered around the globe, some joined, some not - Sunstorm makes mention of them, as does Moonfire. They have chased the Nictus "across half the universe."
(Spoiler) Requiem planned to use Shadow Cysts ('Shadow Seeds') planted across the globe to take over and make Earth a new Nictus homeworld. During the Kheld arcs, you find a 70 year old map showing the plans. So they've been growing and planning this for quite some time. Why they've waited this long? Hard to say. One guess is that it's a combination of an increased Peacebringer presence and technology finally getting to a point where they can force Nictus into unwilling hosts en masse. (Moonfire TF.)
How did they GET here?
Kheldians are able to use a combination of their light form and alien forms to propel themselves great distances at faster-than-light speeds. A forum post quoting Avatea details the mechanism:
This description comes from FloatingFatMan, who requested an explanation from the devs to write some fan fiction, the first part of which illustrates the method of travel described: 
Interestingly, the City of Heroes comic shows the Peacebringer that joined with Horus arriving in a spaceship, but those comics aren't exactly canon (for instance, they show the Nova form as the Kheld native form). It is possible that the Kheldian was injured an unable to travel the ordinary way or that it had other items in the ship that it couldn't carry with it while in light form.
Although it can be assumed that Nictus are able to travel in a similar manner, they can also be transported within Shadow Cysts. The following is pure conjecture: Nictus can travel great distances very quickly, by using a Shadow Cyst. The Cysts are made by (or from) weakened Nictus, and are quite tough. It is plausible that a colony of Nictus, weakened, became encysted and scattered like dandelion seeds, with one landing on Earth some millenia ago.
Creating Cysts requires three things - a weakened Nictus, the "right" crystals, and the "right" energy available. This leads to some of the speculation at the end of the guide - that the reason there was no earlier attempt to take over Earth is that we were too primitive until recently. Now, though, more cysts could be made and a takeover could be contemplated.
Not Quite Kheldians
Let's break these into three levels:
Quantums are nothing more than your average bad guy who is armed with a Quantum Array Gun. These smack Kheldians rather hard, but other than that, they're just a thug.
Galaxy troops are soldiers who have been infused with Nictus fragments ("pieces" of dying or weakened Nictus, though it seems healthy ones can create them too.) They are given some Warshade-like attacks (such as Ebon Bolt and Gravimetric Snare.) Not all of them are volunteers.
Galaxy troops can opt to go for a "full conversion" after a while to a full Nictus - most likely, given other in game comments, because the shards integrate more fully into the host, get stronger, and keep making the suggestion (plus the "longing for space" that's mentioned.) It's mentioned in a story arc that the conversion rate for Galaxy troops to full Nictus is reaching 40%.
Void Hunter mercenaries are the real baddies, as far as Kheldians are concerned. They're infused with far more Nictus fragments which have been modified with Quantum energy and are out hunting Khelds. They have a Quantum Array gun, as well as additional Nictus damage when in melee that affects the Kheldian inside the host. Their altered fragments provide them with energy resistance, instead of the attacks the Galaxy troops get.
Arakhn's in charge of the Void Hunter mercenaries, though Requiem's secretly been sending "loyalists" to undergo the process. (Shadow Science.)
Two other types of creatures come up in conversations about these - War Wolves and Vampyrii. Failed Galaxy conversions can create War Wolves, though there is also some separate process geared specifically towards making them. Vampyrii are made in a completely separate process that has nothing to do with Nictus shards or Kheldians at all.
There is a portion of the Kheldian story arcs that has you dealing with "shadow" enemies - Shadow freaks, Shadow Vampyrii and the like. These are Freaks and Vampyrii that have been infused with Nictus fragments. Nosferatu himself was considering it. But the fragments have NOTHING to do with the original creation of Vampyrii.
We don't know how Khelds reproduce. Nictus, however, have a means apparently available to them (hinted at in the Galaxy soldier section.) In order to make a Galaxy soldier, N-fragments ("pieces" of a Nictus) are implanted. We don't know how many pieces it takes, or how many you can get from a dying Nictus. 40% of those troops (at this point) opt to become full Nictus. If you can get more than one set of shards from a Nictus - and given there's a bit about creating them from healthy Nictus, as well - you can have multiple Nictus created from one. How much of the original Nictus personality remains is unknown.
In-game, the Peacebringers seem surprised by this, so they most likely have some other means available to them.
We don't know of "native" Kheld gender. The hosts obviously have gender, but native Khelds - unknown.
Of course, part of the fun is naming your character. There are no hard-and-fast rules about Kheldian naming. Everything I thought I found for Peacebringers had some exception. I THOUGHT the name would be "solar object + light reference" but that's broken by Sunstorm, Moonbow and Freeflight... so go nuts. Warshades SEEM to have a "shadow" reference to them - but people seem to go that way (shadow/dark) with them as it is, given their dark coloration in-form.
The names, by the way, are the names of the joined beings. Shadowstar's daughter Lilian is slated to become "the new" Shadowstar, so it's an inherited name or title.
Native Kheldian names? We know of some - Altered Umbra, Perihelion, and Dirge of Entropy. The best "explanation" for these names is it's the best fit our minds and language can come up with for Kheldian names. They are, after all, energy beings.
Some names from within the game:
Peacebringer: Sunstorm (Kheld named Perihelion,) Freeflight, Moonfire, Coldstar, Fullalbedo, Moonbow, Brightshift
Warshade: Shadowstar, Shadowcatcher (Nictus named Altered Umbra), Romulus
Nictus: Requiem (Nictus named Dirge of Entropy,) Arakhn
So just pick something that sounds good to you.
Other bits and pieces, speculation and references
Don't take any of the below as canon. It's just loose speculation or interesting tidbits.
- Requiem is called Science Lord in the original COH comic. Again, though - not canon.
- Earth was either not interesting enough, not advanced enough, or not needed for takeover earlier. Given "recent" advances (Nictus takeover map = 70 years old, Industrial Revolution = 1800s, now up to nuclear power, quantum physics, and space travel) we may have just gotten VERY interesting or much more useful. Nictus may even be (in game, obviously) a driving force behind some advances. Nothing in game to back this up other than needing the "right" energy and crystals for a cyst. It's just an interesting choice time-wise.
- The closest star is 4.2 light years away. Unless Peacebringers have hosts that can survive interstellar space, something Cyst-like or can 'hibernate' as well, it would take nearly half a Kheldian's natural life to get here (assuming that, as an energy being and having no mass, they could travel at the speed of light.) On the other hand, traveling at very near the speed of light results in time dilation effects that shorten the trip significantly for the traveler. Since lifespan is a subjective matter in most cases, a Kheldian traveling in natural form at or near the speed of light may not age during the trip.
- Shadowstar's been mentioned in "ancient Egypt," which gives us 2000-5000 years to work with. Board speculation (or bored speculation?) came up with an interesting idea - that Egyptian funeral rites could have been influenced by experience with the Nictus, with the Pyramids as a hiding point for a Cyst. Nothing in game supporting it, but it's an interesting idea to play with. (Fuzun.)
Most of the "in game" quotation was made easier by having http://www.nofuture.org.uk to refer to, on my part, as well as being able to go back and read through story arcs.
You can, if you don't mind spoilers, go to http://www.nofuture.org.uk and read through the Moonfire, Sunstorm, Shadowstar and general Kheldian arc for information, as well as the Issue 3 introduction of Kheldians over at the City of Heroes homepage.