According to the NFL Collective Bargaining Agreement (CBA), the waiver system is a procedure by which player contracts or NFL rights to players are made available by a club to other clubs in the League.
When a player is placed on waiver, the team holding the rights offers the contract to any of the 31 clubs, who either put in a claim for the contract or pass, or waive their right to the players services. The selection process during the season is based on the inverse of won-and-lost standing of that week, similar to the formula used to determine the NFL draft order. [As per KAISER TORO's post: This waiver wire order is reset on September 28th to reflect the current worst win/loss records in the league.]
Between July 4 and the conclusion of the regular season, the waiver system takes 24 hours to complete, whereas in the off-season, its 72 hours.
If all 31 teams pass on a players rights, his contract is officially terminated, and he becomes an unrestricted free agent regardless of the number of accrued seasons he has. However, if hes picked up by a new team, his original contract details carries over to the team acquiring his rights; the exception being the signing bonus, which is the responsibility of the team that originally held the players rights.
All waivers are no recall and no withdrawal, meaning a team cannot change its mind once it makes its decision.
Further, if a team claims a player off waivers, the team releasing his rights is not entitled to any form of compensation, as is the case in free agency when a restricted free agent is signed to an offer sheet.