The buy-in is the amount that a player must pay to enter a game. In a cash game, it is the amount he must bring to the table.
In a tournament, it is the amount he must pay into the prize pool. The whole amount paid to enter the tournament is the buy-in plus the entry fee, where the entry fee goes directly to the host of the tournament. A tournament advertised as "$10+1" means that a player pays 11 dollars in all to enter the tournament, 10 for the prize pool and 1 to the host.