Waivers just might be the most complicated single aspect of the rules. In the rule book, a waiver is defined as "... a permission granted for certain assignments of player contracts or for the unconditional release of a Major League player ..."
If a player placed on Major League waivers is not claimed by another team during the three business days after waivers have been requested, then the players is said to have "cleared waivers," and the team has secured waivers for the remainder of the waiver period.
And what does that mean? Essentially, the team can do with the player's contract as it pleases. This generally means one of three things:
(1) They can send him to the minors (subject to his consent, if he's a "Veteran Player," more on that below).
(2) They can release him, which makes the player a free agent and thus available to sign with any team.
(3) They can trade him to another team, even if the so-called "trading deadline" has passed. Any trades made after July 31 may only involve players who have cleared waivers.
If a player doesn't clear waivers -- in other words, if he's claimed by another team or teams -- the club requesting waivers may withdraw the waiver request.
If the club doesn't withdraw the waiver request, the player's contract is assigned in the following manner:
(A) If only one claim is entered, the player's contract is assigned to that claiming club.
(B) If more than one club in the same league makes claims, the club currently lower in the standings gets the player.
© If clubs in both leagues claim the player, preference shall always go to the club in the same league as the club requesting waivers.
There are other, more esoteric rules involved here. For example, during the first 30 days of the season, the previous season's final standings are used to determine claim order, rather than the current standings.