Drafting is a game mechanic in which players take turns choosing items, usually cards, from a shared pool.
Drafting is a common way to set up games; for example, in collectable card games such as Magic: The Gathering, players may each open a deck, choose one card from it, and pass the deck to the player next to them. In this way they decks circle the table, each player taking the cards that they find most valuable. This is seen as a fair way of allowing players to build the deck that they want, usually with the added benefit that they keep the cards that they draft... meaning that there is the additional metagame that they may in fact be collecting some cards not for the upcoming game but for building their decks for future constructed tournaments.
Many games use this mechanic as a part of their game play, including Seasons, Splendor ('drafting' gems), SmallWorld, Agricola, and Ticket to Ride, but these would not generally be referred to as drafting games.
Likewise, most deckbuilders are based on a form of drafting, where all resources are bought from a shared central pool; this is not pure drafting, however, as players do not have equal opportunity throughout the game to acquire cards -- in fact, a central goal of such games are to improve your ability to buy these cards more quickly than your opponents. (These games include Ascension, Star Realms, Splendor, Puzzle Strike, and many others).
There is a much smaller category of games in which drafting is the primary mechanic; starting decks are passed around, each player chooses one card per pass, and at the end the resultant combinations are totalled up. It's more fun than it sounds, at least when it is done right. Games that are primarily drafting games include 7 Wonders, Sushi Go!, Biblios (first half drafting, second half bidding), Treasure Hunter, Fairy Tale, Nevermore, and Tides of Time, among others. Good drafting games tend to have variable starting conditions among players, multiple possible successful strategies, and variable decks (usually accomplished by simply holding some cards out from the master deck for each game), to help keep a limited set of cards interesting and variable over multiple games.
If you are interested in trying a drafting game, Sushi Go is quick, fun, and easy. 7 Wonders is not as quick and easy, but is very popular and is no more difficult to pick up than the average board game.