General term for the technology that allows games to be replayed?

The most common term used for this is simply replay.

Game replays are commonly used in the prevention and detection of cheating, especially when used in electronic sports. Game replays are often watched to develop skill at a game, by watching better players and their techniques and build orders. Replays are also used to raise stature in a particular game's community, and entire websites are dedicated to the replays.

Some games (primarily the Counter-Strike series) also use the term demo.

One of the most convenient features of the entire Counter Strike franchise is the demo viewer. This is no different in the first-person shooter’s latest edition, CS:GO. The demo viewer is handy for beginners and professionals alike by allowing them to examine their performance.

And how a replay is stored depends on the game in question. Often they're just plain or encrypted text files with a unique file extension.