Game playing programs work by analyzing millions of positions that could arise in the next few moves of the game. Typically, these programs employ strategies resembling depth first search, which means that they do not keep track of all the positions analyzed so far. In many games, it is possible to reach a given position in more than one way. These are called transpositions. In chess, for example, the sequence of moves 1. d4 Nf6 2. c4 g6 has 4 possible transpositions, since either player may swap their move order. In general, after n moves, the maximum number of possible transpositions is n!². Although many of these will be illegal move sequences, it is still likely that the program will end up analyzing the same position several times.
To avoid this problem, transposition tables are used. Such a table is a hash table of each of the positions analyzed so far up to a certain depth. On encountering a new position, the program checks the table to see if the position has already been analyzed (this can be done in constant time, without searching through the entire table). If so, the table contains the value that was previously assigned to this position; this value is used directly. If not, the value is computed and the new position is entered into the hash table.
It must be noted that the computation saved by a transposition table lookup is not just the evaluation of a single position - if that were the case, it would hardly be worth the effort, since evaluation functions are designed to be very fast anyway. Instead, the evaluation of an entire subtree is avoided. Thus, transposition table entries for nodes at a lower depth in the game tree are more valuable (since the size of the subtree rooted at such a node is larger) and are therefore given more importance when the table fills up and some entries must be discarded.
The hash table implementing the transposition table can have uses other than finding transpositions. In alpha-beta pruning, the search is fastest (in fact, optimal) when the child of a node corresponding to the best move is always considered first. Of course we have no way of knowing the best move, but when we use iterative deepening, the move which was found to be the best in a shallower search is a good approximation. Therefore we try this first. For storing the best child of a node, we use the entry corresponding to that node in the transposition table.