How it gets called
The 1st AD knows from the shot list which setup is last. Once the second-to-last setup (the Abby Singer) is done, the AD announces the next one on set and on all walkie channels:
"This is the martini. Sc. 45, shot C, Maya's close-up at the car door. Martini shot, everyone."
Departments that haven't already begun packing now start in earnest. Video village begins to shut down non-essential monitors, grip starts wrapping carts that won't be used, and the 2nd AD finishes the production report times so wrap can be logged the moment the take is approved.
What happens right after
When the director and script supervisor are satisfied, the AD calls "Check the gate" (on digital, the DIT or 2nd AC confirms media is good) and then "That's a wrap." Camera wrap time goes on the daily production report. Cast are released, and their dismissal times go on the Exhibit G.
Martini shots that multiply
A day on a fictional feature is planned to wrap at 7:00P with 5 2/8 pages. At 6:40P the AD calls the martini on Maya's close-up. The director watches playback and asks for an insert of her hand on the keys. The AD has to decide on the spot: is there time, and does adding a shot push the unit into overtime or past the location's permit window? If yes, the insert moves to a later day on the pickup shot list. If no, the AD announces "New martini: insert, hand on keys" and the crew stops wrapping. Each extra "martini" costs time for everyone who had already started to pack.
Confusions
People new to sets sometimes call the Abby Singer "the martini" and vice versa. The order is: Abby Singer, then martini, then wrap. The Abby Singer is named after assistant director and production manager Abby Singer, who announced the second-to-last shot so crews could get ready for moves (Wikipedia).
If the day includes a company move, the last shot at the first location isn't a martini, since the day continues. ADs usually call that one "last shot here" instead.