Salvaging is the term used to refer to executions in the Republic of Gilead. Those who are executed are referred to as having been 'Salvaged'.
Women's Salvagings
The executions of women are held in public and presided over by the Aunts. Handmaids have to sit in the center of a lawn near the stage where the 'Salvagings' take place and are forced to participate. Women criminals are usually hanged. Women can executed for crimes such as adultery or the murder of Handmaids. It is implied that women who are to be executed are drugged beforehand to keep them from causing trouble. Handmaids, and possibly other women in attendance, are not given breakfast prior to attending a Salvaging, which is done from practicality, as the sight can cause the women to throw up.
Handmaids are given the rope used to hang the condemned and pull altogether, thus sharing in the 'duty' of executing those who commit crimes against the regime. Offred states that the Aunts used to announce for what crimes the women are being executed for; however, they stopped doing this due to the fact it seemed to encourage a spate of similar crimes.
Men's Salvagings
Little is known about men's Salvagings, as women are not allowed to attend; however, after being executed, criminals's bodies are displayed on the Wall as further punishment and to serve as a warning. Men are often executed for 'Gender Treachery' or for providing abortion or birth control in the pre-Gilead days. An executed priest is also seen hanging on the Wall at one point, due to his beliefs conflicting with those of Gilead.
Rarely, male criminals are subjected to 'Particicutions', where a man is literally torn or beaten to death by Handmaids. Often, men executed this way have committed crimes such as rape, which are deeply personal to the women; however, Ofglen mentions that one alleged rapist was actually a member of Mayday, the underground resistance, thus suggesting the government executes traitors and rebels on trumped up charges to remove them as a threat. It's highly possible they may do this with both men and women.