Eldrazi
The Eldrazi hunger without limit and consume without pause. —Oblivion Sower flavor text
Eldrazi are (mindless?) lovecraftian abominations with an appetite that makes Tyranids look anorexic. When Tyranids are through with a world there is at least a barren piece of rock. They also make tyranids seem like natives to wherever they are attacking. These cosmic horrors are so alien they don't even truly enter the plane they feed on. They have their (physical) shadows go in. Then the shadows go super organism, creating swarms of brood that devour the plane of mana and life energy while the actual Eldrazi Titans (e.i. the actual Eldrazi) just chill in their home, the Blind Eternities[1]. These shadows are represented in game as creatures, and Rise of the Eldrazi drones aside, colorless creatures. At the time of their release in Rise of the Eldrazi, the only other colorless creatures are artifacts and face down cards. The card Scion of Ugin has since joined that list.
Worshipers
Gods don't die. They merely slumber. —Ayli, Kamsa cleric
The Eldrazi Titans, before they were freed grew to be considered gods, which Nahiri probably inadvertently had a part in. After becoming free and completely fucking up the plane by eating almost everything (mainly through brood) and even just being there, some idiots still worship them and act accordingly.
Kozilek
Kozilek was worshiped under the following names:
- Cosi (merfolk)
- Talib (kor)
Ulamog
Ulamog was worshiped under the following names:
- Ula (merfolk)
- Mangeni (Kor)
Emrakul
Emrakul was worshiped under the following names:
- Emeria/Em (merfolk)
- Kamsa (kor)
The Titans
Each of the Titans has it's own traits
Kozilek
Kozilek's brood have a bunch of eyes.
Ulamog
Ulamog is the hungriest eldrazi. He and his brood have a bony mask/helmet like structure.
Emerakul
Emerakul's brood has "fleshlike lattice structures, uneasy bilateral symmetry, and tentacles that end in knobby, vestigial digits"
In Game
First off, players, and WotC, generally only call straight up Eldrazi (as in no other creature subtypes) "Eldrazi", so thats what we are gonna do here. The Eldrazi are fatties, the weakest one is a 7/7 (and by the way, thats a common). Thematically you are supposed to get Eldrazi Spawn out before you get out your Eldrazi, to help (or flat out entirely) pay for Eldrazi. This is (in at least some formats) entirely viable. Eldrazi spawn are 0/1 creature tokens that you can sac to gain 1 colorless mana. Cards that make these tokens generally do something else, so the token is basically a rider for mana/the chumpest of chumps for later use.
Rise of the Eldrazi
Annihilator
Every Eldrazi released so far has the Annihilator keyword ability (once Battle for Zendikar comes out, this will no longer be true). When a creature with a Annihilator attacks, the defending player sacs a specific number of permanents, because, you know, the fucking plane is falling apart. Annihilator is a very scary keyword ability, so in a multiplayer game its probably wise to stay away from having it unless you're okay with getting ganged up on by EVERYBODY! Between annihilator and Eldrazi Spawn, its probably a good idea to have things that benefit when something gets sacrificed, like Mortician Beetle (which, you may have noticed came out at the same time as the Eldrazi tribal cards). After the release of Rise of the Eldrazi, Wizard's R&D then looked at the market research and decreed it was: NOT FUN. So we probably won't see new cards with it for a long time (if you exclude reprints), if ever.
Eldrazi Titans
If one of the 3 Eldrazi Titans ends up in your graveyard, you shuffle your entire graveyard into your library. So Eldrazi Titans can be used as mythic rare anti-library-to-graveyard mill tech when the format allows. If a deck is designed to cheat a creature into play, that creature is generally chosen to be Emrakul. Even though its first 2 abilities are only helpful if you CAST it, its so powerful that decks that cheat creatures into play without casting them still usually choose Emrakul (who cost 15 is a 15/15, immune to coloured magic, crazy fucking monster) (if feasible, you can't sorcery speed reanimate any of the Eldrazi Titans).
Battle for Zendikar
The Duel Decks: Zendikar vs. Eldrazi announcement revealed that an Eldrazi was being printed without Annihilator. The Eldrazi have a exiling your opponents library theme. For the most part, it isn't efficient enough to warrant a milling strategy. Instead it acts as an enabler for another Eldrazi theme: Removing cards an opponents cards from exile and putting them in the graveyard for some sort of benefit. Lastly the Eldrazi Spawn theme is back except they are 1/1s and called Eldrazi Scions now.
Devoid
Devoid is a static ability that makes a card colorless in all zones despite what it's mana cost would tell you. Devoid creatures sometimes have activated abilities that use the same color as their colors in their mana cost. Devoid does not impact a card's color identity in commander. This is R&D's way of giving the Eldrazi abilities that are assigned to certain factions of the color pie (e.g. direct damage and red), in a fashion that denies access to those abilities to other colors, without giving (or at the very least down playing) the Eldrazi that color's flavor. Why not just use a color indicator instead of an ability word? Well MaRo initially wanted it to be based on the card frame. The rules team said they couldn't do that. So they had someone do a colorless color indicator. She couldn't make it so players picked up on it. Communication is important, so they resorted to having an ability word.