It is a tournament where there will be at least $1000 in prize money distributed amongst the entrants.
If very few people enter the tournament, there can be a large "overlay", where the casino has to put up its own money to pay out the entrants.
If the buy in is $50 and the tournament has 10 entrants, then there would normally be $500 in prize money. This means that the casino is putting an additional $500 of their own money in the tournament. If there are 30 entrants, then there is already $1500 in prize money, so the casino doesn't need to put up any additional money.
Poker Stars and Full Tilt have guarantees that can be bigger than $1,000,000 during their FTOPS and other poker events.
This is a marketing tactic to ensure players that registering for a tournament is worth their while.
On Sunday (Mother's Day) Full Tilt Poker had a $1.5M Guarantee which had an overlay of more than $250,000.