A hand history is a transcript of a poker hand, usually issued electronically from an online poker site. Generally, a person can only see hand histories for hands in which they were involved as a player (e.g. were dealt into the hand), and the histories contain only the information the player could have seen at the time of the hand. For example, facts like how much each person bet, who folded, what the flop was, and so on (facts which would be observable by anyone watching the actual hand in play) are recorded in the hand history.
Often hand histories also include a listing of the hole cards of every player who still had a live hand at showdown, even if that player chose to muck their hand rather than show it. This simulates the I Want To See That Hand rule which is practiced by most live cardrooms and casinos.
Some online poker sites provide hand histories automatically, storing them on the player's local hard drive (e.g. PartyPoker), while others provide limited hand history reporting (e.g. Planet Poker's hand history searching facilities) or none at all.