Waivers are the process by which a released NFL player is offered to every other team before becoming a free agent. Clubs claim in a priority order based on record, worst first, and the highest priority claim wins. Players with enough accrued seasons skip waivers and become free agents immediately.
A claim is not a bid. The claiming team takes on the player's existing contract as it stands, so a team is deciding whether the player is worth the money already written down rather than making an offer of its own. That is why expensive players often clear waivers and cheap young ones do not.
The accrued seasons rule is what separates the two paths. A player who has been in the league long enough, released outside the part of the season where waivers apply to everyone, becomes a free agent straight away and can choose where to sign, which is a considerable advantage over being claimed.
Waivers also explain the annual roster cutdown. When every team trims to its final roster on the same day, hundreds of players hit waivers at once, and the worst teams from the previous season get first look at all of them.