Particicutions are a specific type of execution or 'Salvaging' in the Republic of Gilead. It involves the execution being carried out by a group of Handmaids, who are usually required to attend Salvagings. The word itself - Particicution - is a portmanteau of the words 'participation' and 'execution'.
Usually, those executed in Particicutions are men convicted of crimes such as rape or other forms of violence against Handmaids, as this makes it more personal for the Handmaids.
In the Particicution featured in the novel, the condemned man is accused of raping a pregnant Handmaid, resulting in miscarriage. That being said, it is revealed that the man was in fact a member of the Mayday resistance, suggesting that the government allows traitors and rebels to be executed on false charges, so as to incite the Handmaids to violence.
During the Particicution, Handmaids are allowed to use any means they wish to kill the condemned, though as they are not permitted weapons, they must use their bare hands. The condemned is often beaten to death or torn to pieces by the Handmaids. They are encouraged to drive themselves into a frenzy and it is implied that Handmaids use Particicutions as an outlet for their pent up rage and grief over their loss of freedom and rights. Particicutions are apparently uncommon, as Offred had only ever attended one.
In the novel and film versions, the Particicution happens directly after a women's Salvaging and is therefore watched by the Marthas, Wives and Econowives present; in the TV series, it is a closed event and only Handmaids attend it.
According to the novel's epilogue, Professor Pixieto--leading a conference on the study of Gilead--states that his research yielded the fact that Commander Waterford was the mastermind behind the invention of Particicutions.
Particicution scene from the 1990 film adaptation of The Handmaid's Tale: