Veteran players are cut, while young players are waived.
When a veteran player is cut, then he's cut and able to sign with any team.
When a young players is cut, then he's not technically cut, but waived, because he goes on the waiver wire for a 48 hours periode. Any team can make a claim for that player, but has to be willing to pick up his current contract as well. The claim is made at the league office and if more teams are claiming a player then the worst team is awarded the player (just like the draft)
If nobody claims the player during the 48 hour peirode, the he's a 100% FA and is not garanteed his old contract.