It depends on the sport. For the NHL, cut is when a player is let go from training camp. When a player is waived, he is put on the "waiver wire" where any team can put a claim on him before he is sent to the minors. Teams have 24 hours to put in a claim. If more than one team claims a player, the team lower in the standings gets the player. A buy-out is when the player is paid a certain amount of money over twice the remaining length of the contract to leave the team. I forgot how buyouts are calculated but teams have to do it...before the free agency period begins I think? Again, I forgot. Anyways, after a buyout, the player becomes a free agent. But that's basically a summary of how that stuff works in the NHL.