A draft refers to when players make exclusionary choices from a common pool; you’ll see this sort of thing in professional sports to determine where players wind up. In card games, however, a draft usually refers to a mechanism where players each are given a handful of cards, choose one card, and remove it from the hand, then pass their hand on to the next player.
Draft based on cards can put complexity on the cards themselves, rather than in the fundamental structure of the game; the game has a really simple rule of Take A Card, Pass Everything Else Along. That means players can focus on just the cards in their hand, that they can select from, and not need to worry about what else is going on at the table (though they might).
Draft is also simultaneous: Players will all be taking their turn at the same time, meaning that even if one player is markedly slower than the others, players won’t be waiting the entirety of their decision making process, since they have to do some of their own decision making process.
Draft is a super duper complicated way to design your game. In the case of a bigger game where draft is used as a component, like Magic: the Gathering there are thousands of words written every month or so about ways to do it well, strategies for repeating the same game, things you can do to handle the variance and things you can do to capitalise on it.
Players get to see a bunch of the cards in front of them; they get to know what they’re passing, and that can mean that players seeking an edge may feel obligated to remember everything going around the table. You can find this a bit paralysing.
You can make drafts open, where everyone makes it clear what they’re taking each turn, which gives players even more information, and may be even more paralysing. If you do this it’s often best to have a common pool that everyone takes one thing from at a time. I’m less fan of this but it can be a good way to make players hold grudges against one another.
Some drafting games include Magic: The Gathering’s limited format (known as draft, helpfully), 7 Wonders, and Inis.