each league has a time that a player takes to clear waivers. when you look at your league homepage there will be the record, points scored and waiver order. if someone is clearing waivers and someone else has priority over them that person can take the player off waivers because they have a higher rank. your record and the amount of transactions you take part in have an effect on your waiver order.
if you cut a player and they haven't cleared waivers yet you cannot pick them back up until they clear waivers and the player is an eligible free agent again.