Simple. When someone drops a player, he goes to that purgatory called 'waivers' where he waits around for someone to claim him. Teams that try to claim him before the waiver period expires (default is two days for yahoo) have to wait until said period expires to find out if they get him. I think that if the team ranked #1 claims him, this automatically ends the waiver period and he goes to that team. The ranking is the opposite of the draft, so if you drafted 5th in a twelve-team league, your waiver ranking is seven.
If no one ranked above your team claims the player, you get that player after the waiver period ends. A team that successfully claims a player via waivers then gets pushed to the end of the line.
If nobody claims that player, he becomes a free agent after the waiver period and then any team can pick him up without waiting.