Hand History in poker is simply a description of a single hand, exactly as it happened, with all known cards, betting actions, and winner(s).
There are a couple of standard ways for writing down hand histories that are recognized in the poker community, and using this notation makes sharing your hands with others much easier.
A properly written hand history will give anyone who looks at it the full breakdown of a hand from preflop round to showdown. It describes every bet and raise and shows all known hole cards and dealt community cards.
Poker hand histories are valuable learning tools. They help players learn from their own mistakes and absorb knowledge from hands played by other players by seeing their stats on the HUDs.