It depends on how long your league has set up the waiver period to be. In my league it is 2 days, so if you dump a player today, no one can add him until Monday.
On your league's "Overview" page, look on the right side under "League Info" and you will see a link for "League Settings." Click it and look for "Waiver Time."
It is a good trick to use, especially if your opponent is weak at the position you are dropping. It effectively removes 2 options from him.