So, if a player in a Public League is dropped on a Wednesday afternoon, all managers can claim him until Friday at 11:59 pm PT. The player will then be 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 details, which can be found via your League Overview page.
If more than one team claims the same player, the team that is highest on the 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 are 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 you pick up a player, but later realize you did not want him on your team, you may immediately waive him. However, you can only claim and waive the same player two times every 24 hours.