Buy the Button
To buy the button is to pay or raise to get position on the button.
The simple explanation
To buy the button has two meanings. In some cash games, a player who has missed the big blind can pay both blinds to play on the button on the next hand instead of waiting. It also describes raising from late position to make the players between you and the button fold, so you can act last.
A closer look
The first meaning comes from a rule used in northern California casinos: a new player may pay both blinds and take the button's place for one hand. The second describes a position-based tactic: if the player on the button folds, you play with an advantage after the flop. Players use the rule to avoid missing an orbit, and the raise to take control of the betting.
A practical example
A new player sits down and buys the button by paying both blinds.
Full explanation
To buy the button is to pay or raise to get position on the button.
To buy the button is to pay a fee in order to play on the button. In some cash games, a player who has missed the big blind can post both blinds, in the small blind position, to be dealt in on the button next hand.
The small blind amount is dead in the pot, since the player does not get the option to raise with it.
The phrase also means raising in late position to push out the players between you and the button, so that you play the rest of the hand with the best position.
Which meaning is intended depends on the context.