I may have complained a bit about what Kill Team lacks, but let's take a look at what is included. There are, as noted before, 16 factions in the game at the time I am writing. Several of these are technically allies with one another, although they do not combine into a single cooperative squad for a "battle-forged" army. There's no reason not to use whatever models you want in a friendly game though. Maybe some Blood Axe Orks agreed to join the Tau to fight the combined threat of Necrons and their Space Marine allies. After all, stranger things have happened in official canon.
First things first: Yes, there are a lot of strange names here. Games Workshop recently decided to rename just about every faction to differentiate their product lines make it easier for their lawyers to trademark stuff. Warhammer 40,000 started as a sci-fi parallel to Warhammer Fantasy Battles, and that game was in turn influenced heavily by major works of fantasy literature. Yes, this means there were once space dwarfs. They were called "Squats." They disappeared early on. Rest in peace, vertically-challenged heavily-armed biker gangs...
The Eldar
Asuryani
Craftworld Eldar
These are essentially "fantasy high elves IN SPACE!!!" Their home planets were destroyed (long story...) and now they live on giant space ships that serve as mobile colonies. They use advanced weapons and tactics, and have an answer to everything anyone else has on the table, but it takes some skill to deploy them well.
Drukhari
Dark Eldar
Evil space elves that thrive on torture and excess as their response to the previously-mentioned long story about their home planets being destroyed. This army is fast and vicious, but although they hit hard, they can't take a beating in return if their assault wasn't enough up front to take out the foe. The epitome of the term "glass cannon."
Harlequins
Elf Murderclowns
These space elves seek to unite the dark elves and the craftworld elves by performing the history of their race (The aforementioned long story, but wait, there's more!) by choreographed pageantry, and killing the assorted threats to their people very, very violently and stylishly.
Necrons
Robot Zombies
An ancient race of generally unpleasant beings who got themselves turned into living metal (another long story...) and now hate all life in the galaxy, but have been taking a long nap for the past few millennia. And they are starting to wake up, invariably on the wrong side of the bed. They carry very effective guns. They have a chance to get back up after being killed through a rule called "reanimation protocols." This rule used to be called, "We'll be back." Whether they speak with a heavy Austrian accent remains to be confirmed.
Orks
Fantasy orcs, but spelled differently
Green. Mean. Big. Not especially bright. English football hooligans that wage interstellar war. They're also so psychically powerful that their ramshackle contraptions work just because they believe they do. They are the comic relief faction in a game full of grim darkness, because they're not trying to save their species, redeem themselves from an old failure, destroy order, or purge all life from the universe. They're in it for the fun! And fun means smashing enemy faces or dumping a lot of ammo in their general direction.
T'au Empire
For the Greater Good!
This race was added to Warhammer 40,000 in 2001, and doesn't really have a fantasy analogue. Depending on who you ask, they're either "weeaboo space commies" or the least grimdark species in the game out to bring sanity to an insane galaxy. In combat, they excel at ranged weaponattacks, but crumple like wet newspaper in close quarters.
Tyranids
The Tyranid Swarms
Bugs in Space. Also, xenomorphs.
The swarm of strange creatures from beyond the galaxy is here to consume all life as it passes through. It adapts the creatures it encounters into new forms that better serve the purposes of the hive mind. Blizzard ripped off this species when they made the Zerg in StarCraft, if that rings a bell.
Genestealer Cults
They're already here...
Genestealers are a sub-faction of the Tyranids that can alter the genes of the species they encounter and develop secret societies that subvert planetary defenses and control the society long before the main swarm arrives. The subverted beings are known as "genestealer cults," because they religiously venerate a twisted vision of the coming swarm. They carry everything from military weapons to repurposed tools into combat, and they have mutations that make them better at fighting than a normal humanoid.
The Imperium of Man
Adeptus Astartes
Space Marines
These guys started out as basically knights in space, but were re-written and expanded in the background to be genetically-engineered super-soldiers who serve the emperor of mankind in his Great Crusade to unite the lost colonies of humanity, and then things went very wrong, and it's been going downhill for 10,000 years. Yup, another long story. And then, a brand new kind of Space Marine was released because the story was finally advancing in the background with the release of a new edition of the game Games Workshop wanted to sell more plastic crack. No, I am not a fan of the Primaris Space Marines story or rules. The Intercessor models do look pretty darn cool though, I must admit. At any rate, this army list in Kill Team covers the basics for all the different chapters of Space Marines despite their differences in the larger games.
Deathwatch
The special forces of the special forces
The Deathwatch is a company of Space Marines sent from all the various chapters of Space Marines for special cooperative missions. They get all the coolest wargear and the best armor to help them take on nigh-suicidal missions against the toughest aliens or literal demons. And their squads are already called Kill Teams, so they are the perfect faction for this game. Build 20 different loadouts, and pick the best 5-7 for your 100-point force against any given foe!
Grey Knights
This is a nonexistant special Space Marine force composed of psychically-gifted super-solders whose primary duty is waging war against demons that don't even exist, so seriously, stop asking questions, citizen. Even if they did exist, they would not be overpowered combat monsters suffering only by a lack of numbers. Now please report to your local commissar for re-education.
Astra Militarum
Astra Mili-what? You're in the Imperial Guard, son!
Are you a standard human with no genetic engineering or cybernetic enhancements to speak of? Here's a serial number, a lasgun, some basic training (if you're lucky) and a ticket to the nearest warzone. You'll probably die long before promotion is an option, but you're the best line of defense we've got! Besides, if you shoot the enemy enough, they have to die eventually! And no matter how much I try to poke fun at the Guard like this, Games Workshop already covered it even more in-universe!
Adeptus Mechanicus
The Machine Cult
Humanity has gone backwards technologically over the past 10,000 years, and what technology remains is treated with literal religious veneration. It's another long story. The Machine Cult holds the secret knowledge to make and maintain the machines that keep the Imperium of Man functioning. They carry special weapons unique to their faction, and have cybernetic enhancements to make themselves more like the perfection they see in machines. They're a lot more capable than the Guard.
Chaos
Heretic Astartes
Evil Space Marines
These former heroes of the Imperium were swayed by the schemes of chaos gods, and now seek to lay waste to the empire they once defended and slaughter the people they once protected. It's also a long story. If the Space Marines are knights, these guys are more like Vikings, or perhaps like the army of the feudal lords seeking to overthrow the king. They are accompanied by cultists who worship the chaos gods and seek to earn their favor on the battlefield. These represent any basic faction of renegade or chaos-aligned Space Marines, and the cultist rules can be adapted for heretic Guard, too.
Death Guard
The March of Entropy
Some Space Marines who follow Nurgle, the chaos god of disease and decay, are blessed with his gifts of disease and mutation. In return, they are supernaturally toughened by their immunity to pain and mutant resilience. Do you want a slow but nigh-unstoppable juggernaut of death? If so, these are your army!
Thousand Sons
ALL IS DUST!
Another special chapter of Chaos Space Marines, these are a legion that succumbed to Tzeentch, the god of change, subterfuge, curiosity, ambition, and dickishness. The results were exactly what should have been expected, and they're all now suits of armor filled with the ashes of their bodies and the angry souls of the marines trapped in a painful state of limbo. It's a long story. But they're all capable of psychic powers in addition to being still effective space marines, so they have that going for them.
If you want to learn more about the many, many long stories behind these factions, you have two options. The first is Lexicanum, a wiki where fans have compiled as much official information as possible to create as complete a history of the galaxy as possible based on all the official sources available. The other is 1d4chan, a tongue-in-cheek wiki based on the shenanigans of the 4chan /tg/ (traditional games, as opposed to vidya games) board that frequently employs profanity and occasional NSFW imagery along with lots of inside jokes. Choose wisely.