An optimisation of the minimax algorithm for choosing the next move in a two-player game. The position after each move is assigned a value. The larger this value, the better the position is for me. Thus, I will choose moves with maximum value and you will choose moves with minimum value (for me).
If it is my move and I have already found one move M with value alpha then I am only interested in other moves with value greater than alpha. I now consider another of my possible moves, M', to which you could reply with a move with value beta. I know that you would only make a different reply if it had a value less than beta. If beta is already less than alpha then M' is definitely worth less than M so I can reject it without considering any other replies you might make.
The same reasoning applies when considering my replies to your reply. An alpha cutoff is when your reply gives a lower value than the current maximum (alpha) and a beta cutoff is when my reply to your reply gives a higher value than the current minimum value of your reply (beta).