The Budapest Gambit is one of the least common gambits but still offers black a lot of interesting play. In the main line for example, white can easily fall into a trap that ends in checkmate very early in the game.
In the second move, black looks to give up his pawn on e5 but then starts to develop of his pieces to add pressure on the e5 pawn. White will be unable to hold onto the extra pawn and therefore many times will just give back the pawn and continue to develop pieces and not worry about the pawn advantage.
In most gambits, the side giving up material will dictate how the game continues, in the budapest gambit it is usually white that decides how the game will continue. In the main line it is white that can decide whether to stay up in material and have doubled pawns or to give back the pawn and have a double bishop pair. This is not to say it is still not playable for black, but white does have more options in the budapest than in other gambits.
Watch the video below to get a detailed explanation of the Budapest Gambit.