Tedomari means the last play. It is used not only for the last play of the game, but for the last play at different stages, and for the last play before the size of plays (temperature) makes a significant drop. As a general rule you want to try to get tedomari. There is a saying about the last gainful play of the game, tedomari is worth double.
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Two problems by trucid, composer of several excellent problems where the last play is key.
When it's down to mutual damage, you can call it 'who blinks first'. If after some plays each advancing into the opponents' territory, Black has a threat worth t and White has a threat worth t', Black hopes t is worth noticeably more than t'.
Then if it's White's turn, he can choose between (a) stop Black's threat, and so Black's play stopping White's threat gets tedomari, or (b) play the move setting up the threat but allow Black's threat too - White carrying out the threat is tedomari all right but White has already lost too much.
See more in /Discussion.
Tedomari applies in other stages of the game, as mentioned at the start of this page. Here is one from the fuseki. The following position is Black to play and is from the book Jobansen Kono Itte by Fujisawa Shuuko:
Shuuko says that the correct move for Black is at the circled point, which one could call tedomari and the only move.
Further discussion on /Discussion page.