Normally a card’s color is defined by the types of mana required to cast it. Devoid changes this default so that a card can cost colored mana while still being colorless.
This has a number of benefits when combined with various other cards. For example, Eldrazi Temple produces 2 mana, but it can only be used to cast colorless Eldrazi cards or activate abilities of colorless Eldrazi.
Herald of Kozilek has Devoid itself, and also makes other colorless spells cheaper.
Devoid also makes it so that from things with protection from certain colors don’t work. For example Goblin Piledriver has protection from blue. Without Devoid, Ulamog's Reclaimer wouldn’t be able to block the Piledriver.
Eldrazi by theme are colorless; but giving them colored mana in the cost makes them harder to cast, and makes it so that they don’t all fit in all the same decks. Devoid allows these cards to be colorless while still having those restrictions.
Note that it’s not only an advantage. It can hurt you as well; cards such as Honor of the Pure will not work with your Devoid creatures, even if they cost white mana to cast.
See this article about the development of the mechanic, which goes into more detail about the decisions they made.
Answer from GendoIkari on Stack ExchangeSo if the card’s mana value is 5 of any color plus 2 blue, but has devoid, can I cast that card for 7 colorless instead?
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Normally a card’s color is defined by the types of mana required to cast it. Devoid changes this default so that a card can cost colored mana while still being colorless.
This has a number of benefits when combined with various other cards. For example, Eldrazi Temple produces 2 mana, but it can only be used to cast colorless Eldrazi cards or activate abilities of colorless Eldrazi.
Herald of Kozilek has Devoid itself, and also makes other colorless spells cheaper.
Devoid also makes it so that from things with protection from certain colors don’t work. For example Goblin Piledriver has protection from blue. Without Devoid, Ulamog's Reclaimer wouldn’t be able to block the Piledriver.
Eldrazi by theme are colorless; but giving them colored mana in the cost makes them harder to cast, and makes it so that they don’t all fit in all the same decks. Devoid allows these cards to be colorless while still having those restrictions.
Note that it’s not only an advantage. It can hurt you as well; cards such as Honor of the Pure will not work with your Devoid creatures, even if they cost white mana to cast.
See this article about the development of the mechanic, which goes into more detail about the decisions they made.
Lets look at each question one at a time.
What's special about devoid? - Devoid actually has a lot of things special about it. Devoid is what's known as a characteristic defining ability, an ability that sets something about the card differently from the way it is normally set, like how the ability Changeling sets creature types. Characteristic defining abilities are in effect everywhere, on the field, in the hand, in the grave and in the deck, this means that you can find a Devoid card using Eye of Ugin or Ancient Stirrings!
What advantages does it give? - One of the biggest advantages devoid gives is it counters most of the protections in the game. Protection from (color) is the most common form of protection in Magic, Blinding Drone isn't blue because of devoid, so protection from blue doesn't stop it from damaging, blocking or targeting.
Why is it tied to the Eldrazi creature type? - This is a bit of a lore thing, the Eldrazi are destroyers of worlds, they pretty much drain mana from the planes in the multiverse, leaving lifeless worlds behind. The big ones were always colorless, though with the return to Zendikar and the introduction of Devoid, the color has been drained from the smaller ones now too.
What does not having a color mean? - The color of a card is set based on the colors needed to cast that card, so without Devoid, Unnatural Aggression would be green and Ulamog's Reclaimer would be blue. Most of the benefits of being colorless are covered in the other parts, but there are downsides too. Green Sun's Zenith can't find Catacomb Sifter because it isn't green, and Bearer of Silence can be killed by Doom Blade because it isn't black. And of course just needing this Devoid ability is a downside for the colorless Eldrazi, if it has Devoid that means you need to spend colored mana to cast it, other than Devoid (or sunburst or converge), anything that was colorless doesn't care about the colors of mana spent on it, and that means they can go in any deck no matter what colors the rest of the deck uses.