Threshold is a Magic: the Gathering (a trading card game)special ability that came out with the set Odyssey (released October 2001). Basically, a player has threshold if he has 7 or more cards in his graveyard (the place where used spells/dead creatures go). When a player has threshold, some cards (the ones that have “threshold: blahblahblah” printed on them) he has will gain new attributes.
For example, the card Divine Sacrament is an enchantment that says that all white creatures get +1/+1. It also has “threshold: all white creatures get an additional +1/+1”. If the player that controls the Divine Sacrament has 7 or more cards in his graveyard, he has threshold, and so all white creatures get the additional +1/+1, so they get +2/+2.
One of the most random aspects of threshold is that a player has threshold, not a card. In casual play, this doesn’t really matter, but this can be important in professional games (Worlds, Pro Tour, Grand Prix, etc.) where the prize purse can exceed $25,000.
One final note: the other special set-specific ability that came out with Odyssey was Flashback. Maybe I’ll explain this later.