The situation for Death is a bit complicated, but when considering what counts as a 'faction' you should be looking for what actually has 'allegiance' rules. Going into AoS 2nd edition, there are five main allegiances. They are:
Grand Alliance: Death
The Death alliance rules are currently found in the Legions of Nagash book, and will be reprinted, with few to no expected changes, in the 2nd edition big rulebook and probably the 2018 general's handbook as well. Death lets you freely mix any and every death model, and is currently the only way to run old tomb kings stuff.
Unfortunately, Tomb Kings will be getting the 'Warhammer Legends' treatment sooner or later, which will remove them from matched play by dropping their points support, and will remove them from the Death alliance even in open play games (note how WL Dark Elves don't have the 'order' keyword). Additionally, a lot of undead stuff is rather dependant on their individual allegiance rules. Running a plain Death army cuts skeletons and zombies off from graveyard healing, cuts vampires, necromancers, and deathlords off from faction spell lists. And Legions of Nagash already includes basically everything but tomb kings and flesh eaters, and since tomb kings are going away and there's relatively little synergy between flesh eaters and other death units (necromancers can no longer cast damse macabre on mordant units), basically imo 'grand alliance: death' as an allegiance is basically dead, you can safely ignore it.
Flesh Eater Courts
The ghoul faction, ghoul kings and ghouls only. They have a battle tome, which you'll want for some pretty good formations, but it was an early book, so doesn't have allegiance rules. For those, you'll need the general's handbook 2018, releasing along with the 2e stuff this month. They're not the strongest faction, but they have some great fluff in AoS, basically they're all crazy, and see themselves as noble, civilized people.
As I said, they have some decent formations. Apart from that, they lean heavily on small 'courtier' heroes, which replenish nearby units. Unfortunately, small heroes are pretty easy to snipe in this game. Also, the courtiers don't have their own models, you have to convert them from regular troopers, but since AoS uses batch pricing, that leaves the rest of the box king of unusable, which is frustrating.
Flesh Eaters have very limited ally options, but they don't have that much synergy with other undead anyway.
The ghoul kings have summoning command abilities, which were worthless in AoS before now, but should be pretty good in 2e, though they're going to be limited to one use per king per game. And the courtiers will be slightly harder to snipe.
They're an interesting if a touch undersupported faction. If you want to know more about them, search for Age of Nagash's youtube videos on them, as he used to run them in tournaments and has some good overviews for them.
Soulblight
All vampires and bats. Their allegiance rules are in the Legions of Nagash book, along with a halfway decent formation, access to a nice faction spell lore, and named characters (neferata, mannfred, and a new guy who is just the dragon lord with sword and no helmet, but gets some neat rules).
Their only battleline is blood knights, and they get some minor but interesting bloodline rules. Unfortunately, their unit selection is very limited, and somewhat refundant (vargheists and bloodknights serving anout the same role, and you'll already have to field a bunch of blood knights, so...) and leans too heavily on expensive, offensive elites with not enough defensive weight to stick around and hold objectives. Rumored points decrease for blood knights will help, but probably not enough.
A very minimal, very extreme faction. Interesting to play, but hard to win with, and expensive to collect, unless you find an alternative to the official blood knights. If you're willing to field non-vampire battle-line units (dire wolves are pretty good, and in theme for vamps, imo), then Legion of Blood is probably a better way to run this.
Nighthaunt
They're getting a new book with new rules and new units, so it's too early to say much about how they'll play, apart from that, like Flesh Eaters, they seem somewhat dependant on small, short ranged support heroes, which probably isn't a good thing in competitive games. Their caster, spirit host, and hordy core unit should all be pretty good, but from the battle reports we've seen, their wraithy elite units seem decidedly underwhelming.
The models look great, though, and they're one of the starter box factions in 2e, so at the very least Nighthaunt should be easy to collect and fun to paint.
Most of their units will also be available to Legions of Nagash armies, so if you don't like the Nighthaunt rules and named heroes, you can run them with LoN rules instead. Considering how good the LoN rules are, this is likely to be a better competitive choice, but without seeing the full nighthaunt rules its too soon to say. Still, the option is nice, as a mostly nighthaunt collection will have at least six sets of allegiance rules to choose from. Even if only half of them are decent, that's still a lot of gameplay variety out of the same models.
Legions of Nagash
All current death units fall under the legions of Nagash except for Flesh Eaters, Tomb Kings, and the forgeworld Mournghul. Yes, there are a bunch of mini sub-factions like deathrattle, dead walkers, deathmages, etc etc, but they aren't really playable as stand alone factions, and all are subsumed within the actually playable Legions of Nagash. Supposedly, most of the new Nighthaunt units will also be playable in Legion of Nagash armies. As such, the legions are basically the replacement for the now largely defunct catch-all Death allegiance.
The legions are the strongest and most diverse current death faction by a country mile, set to only get stronger in 2nd edition. The legions are actually a set of four different allegiance rulesets, each themed to one of our named characters. Each of these allegiances shares access to basically the same wide variety of undead units (Nagash can't be fielded in the mortarch allegiances, otherwise everything can field everything), and share a few key rules.
Most prominent of the shared rules are access to two great faction spell lores (one for necromancers, one for vampires, with deathlords having access to both), and gravesites. These are four markers you place on the board after sides are chosen but before deployment. They heal nearby units with the 'summonable' keyword (skeletons, grabe guard, spirit hosts, etc); you can deep strike summonably units from gravesites, provided you have a hero nearby to call them in, instead of deploying them normally; and your general gets a command ability to restore a slaun summonable unit to life via gravesites as well, an ability that won't cost reserve points in 2e and us potentially quite game breaking. I wouldn't be shocked to see errata nerf that one, but even without it, gravesites give your army significant recursion potential that usn't character dependant and allow for tricky deployment shenanigans that significantly offset the slow speed of some of our units.
Grand Host of Nagash
As for how the legions differ, Nagash's legion buffs morghasts and makes them battleline if nagash is your general. Grave guard become battleline regardless, and the command traits and artefacts offer several ways to buff skeletal units. This is the legion if you want to run maxed out, buffed up skeleton hordes supported by powerful morghast hammers. It's also the only legion that can field nagash, who gets to choose three spells from the new lores (other casters only pick one spell). Nagash also has access to a formation that lets him pass wounds off to a morghast bodyguard unit, making him insanely durable, even for his mammoth points cost. And the new edition is set to make him even more dangerous via the new remains in play spells, one of which lets him cast Hand of Dust at a target 20" away.
Legion of Sacrament
Arkhan's legion is all about magic, with a boost to casting rolls, and artifacts and command traits that support or enhance your wizards. The legion was lacking a bit of punch before now, but the endless spells should help considerably. Arkhan also has a good formation that allows him and a coven of necromancers to cast extra spells per turn as long as they stay near a mortis engine.
Legion of Night
Mannfred's legion is all about dirty tricks, with its most notable trait being the ability to outflank a few units, deploying them from any board edge during the game. This is especially deadly with terrorgheists & banshees, which can scream the turn they deploy, or morghast harbingers, which have a good chance of making a charge. Some of the artifacts are quite nice as well. Mannfred's formation isn't too good, but the outflanking ability alone is pure gold, its value really cannot be overstated.
Legion of Blood
Neferata's legion emphasizes the regal vampyric lords of the dead with more powerful blood knights and vampire lords (including dragon lords) than even soulblight has access to. The legion's units also impose and extra leadership penalty on nearby enemy units, enhancing the power of terrorgheists and banshees. Neferata's formation isn't that great, but her legion probably makes the best use of the soulblight formation of outflanking blood knights.
Again, all these legions have access to basically the same unit selection, so you can collect a single army and then field it in very different ways from game to game, or you can commit to a particular legion and collect the specific units that take best advantage of that legion's traits.