The Vintage format allows the use of all cards in the magic cardpool, barring those found in the Unglued/Unhinged sets, and those found on the banned and restricted list. This list can be found on the link provided above.
A banned card is a card that cannot be played at all. In vintage, only "athletic" cards, cards which require you to perform some sort of physical feat like flipping a card or spinning a card, and cards that deal with the "ante" are banned. Ante was how magic was originally played, with players essentially playing each other for a card in the other's deck. The only other card that is banned from the format is Shahrazad.
If a card is restricted, a deck may only contain one copy of it between the maindeck and the sideboard. These are cards that are either format-warping, incredibly strong, or as recently declared by Aaron Forsythe in an article, "un-fun".