Flashback is one of two set-specific abilities of Magic: the Gathering that came on the cards of Odyssey. (The other is threshold). Usually, spells that enter your graveyard stay there for good: they don’t do much other than help give you threshold, and look pretty. However, cards with flashback (they have a little graveyard-symbol in the top left corner) can come back one more time. Cards with flashback will say something like “Flashback: (cost of flashback, usually X mana).” If the flashback cost is paid, the spell will be placed on the stack (basically, cast) again. After the spell does it’s thing, it can’t be flashed-back again. Instead of going into the graveyard, it is removed from the game.
An example of a card with flashback is Firebolt. Firebolt is a sorcery (a type of spell) that says “deals 2 damage to target creature or player.” It’s flashback cost is 4 colorless mana and 1 red mana. If Firebolt happens to be in the graveyard (meaning it’s been cast normally, or just happens to get there), the player who’s graveyard Firebolt is in can pay the flashback cost. If he does, he gets to deal 2 damage, just as if the spell was cast the boring old way! However, after this, the spell is removed from the game, meaning it can’t be flashed-back again.
The predecessor to flashback could be buyback, an ability that came with the Rath Cycle group of MtG sets. Buyback is similar, except the card can basically be flashed-back infinitely, provided the mana is paid.
One last note: you may have noticed that cards with flashback have zero synergy with cards with threshold. Flashback basically removes cards from a player’s graveyard, which would otherwise contribute to the 7 cards in graveyard needed for threshold.