Players can still be traded following the non-waiver Trade Deadline -- which falls at 4 p.m. ET on July 31, except in rare cases -- but they must first clear revocable trade waivers. A player placed on revocable trade waivers can be claimed by any club; waiver priority is determined by reverse standings (worst to best record) in the player's league, followed by reverse standings in the other league.
The player's original club has three options when a waiver claim is placed: It can either work out a standard trade with the claiming club (the two sides have 48 hours to agree to a deal), allow the player -- and all money remaining on his contract -- to go to the claiming club with no return or pull the player back off waivers. A player who is pulled back off waivers can be placed on trade waivers a second time, but at that point the waiver request becomes irrevocable. If a player passes through waivers unclaimed, he can then be traded to any club, free of restriction (though all 40-man-roster players in the trade must clear waivers before being dealt).
Clubs will typically place the vast majority of their Major League roster on waivers once during August, as it can allow them to gauge interest for the upcoming offseason.
August 31 "deadline"
The last day in August is sometimes colloquially (or mistakenly) referred to as the "waiver trade deadline," as players acquired after that date are ineligible to be added to the postseason roster by their new teams.
Teams can still acquire players after Aug. 31, but any new acquisitions from Sept. 1 through the end of the regular season are not eligible to participate in the postseason. For instance: On Sept. 13, 2015, the Blue Jays acquired infielder Darwin Barney from the Dodgers. But because Barney was acquired after Aug. 31, he was ineligible for the postseason roster and did not appear with Toronto in either the ALDS vs. the Rangers or the ALCS vs. the Royals.
Examples of Trade Waivers
The Chicago Cubs claimed Cole Hamels from the Philadelphia Phillies in August 2014, but the two sides weren't able to work out a trade in the allotted 48 hours and Hamels was pulled back by the Phillies.
The White Sox claimed Alex Rios off waivers from the Blue Jays in August 2009, and Toronto elected to allow Rios to go to the White Sox without compensation as a means of shedding the nearly $60 million remaining on his contract.
In August 2013, the Pittsburgh Pirates claimed Justin Morneau off waivers from the Minnesota Twins, and the two sides agreed to a trade that sent Morneau to Pittsburgh in exchange for outfielder Alex Presley and a player to be named later.