All players dropped from a team's roster are immediately placed on waivers. During this waiver period—two days in Public Leagues and anywhere from zero to seven days in Custom Leagues—anyone in the league can claim the player.
So, if a player in a Public League is dropped on a Wednesday afternoon, all managers can claim him until Friday at 11:59 p.m. PT. The player is then assigned to their new team roughly 4-8 hours after the claim deadline.
To find the length of the waiver period in your league, check the League Settings page, which can be found via a link on your League Overview page.
If more than one team claims the same player, the team that is highest on the Waiver Priority List is given the rights to the player. The priority list is initially set based on the reverse of the draft order. However, each time you add a waived player to your roster, you're pushed to the back of the list. This "rolling" list stays in effect for the entire season.
To find your current position in the waiver priority list, go to the Managers List via your League Overview page.
If nobody claims a player during the waiver period, he becomes a free agent. In a Private League with no waiver period, all dropped players automatically become free agents.
If you place a player on waivers, you cannot reclaim that player until after the waiver period has expired. If, however, you decide that you don't want to waive the player, you can cancel the transaction—provided it hasn't yet been processed—by viewing it from your team page and clicking "Cancel."
If you place a claim on more than one waived player at a time, each pending transaction appears on your Team Page along with its current position in your waiver order. All new claims get appended to the end of the list, but you can alter this priority by viewing each pending claim and changing its priority in the box provided.