Magic Formats
There are many different ways to play Magic the Gathering; these methods of play are called formats. There are two broad categories of formats, constructed and limited. In constructed, players build decks before the game from cards in their collections. In limited, players build decks from a limited (hence the name) pool of cards, for example a draft or from sealed packs.
Sanctioned Formats
These formats are sanctioned by Wizards of the Coast.
Constructed
Standard
Standard uses cards from the most recent sets.
Modern
Modern uses cards from 8th edition onward.
Legacy
Legacy uses all magic cards (except for those that are banned).
Vintage
Vintage is the same as legacy, only with a smaller ban list.
Commander
Uses 100 cards decks, with only one of each card being allowed. One card is designated the commander; it is kept separate from the rest of the deck and may be cast at any time.
Limited
Unsanctioned Formats
Constructed
Frontier
A new format that has created a lot of buzz in the community. Uses cards from the M15 set forward.
Kitchen Table
Fun games played between friends. All cards are allowed (unless the players decide otherwise), and players are expected to build decks of comparable power level to the other players, lest they become that guy.