Where it sits in the sequence
Once lighting is done with stand-ins, the 1st AD calls first team back. There's a camera rehearsal, maybe some small lighting tweaks, and then the call:
"Last looks, please. We'll be rolling in two."
The on-set makeup artist steps in to powder shine and fix lipstick. The hair stylist adjusts flyaways that show up under the backlight. The set costumer smooths a collar and checks that the jacket is buttoned the way it was in the master. Props tops off the drink in the actor's glass to the level it was in take one. Then the 1st AD calls "Clear, please, picture's up," and everyone steps behind camera.
On a small indie with one person doing hair and makeup, last looks is the same idea with fewer people.
Why it happens at the last minute
Actors under lights sweat, wardrobe shifts when they sit, and a hair strand can move during a rehearsal. A check done ten minutes earlier might not match what the camera sees now. The key makeup artist and set costumer also watch the monitor during takes and will ask for a quick touch-up between takes if something slips, which is a smaller version of the same call.
Continuity is part of it
Last looks is when departments make sure an actor matches their own earlier takes and setups. The set costumer and hair team keep continuity photos on a phone or tablet for exactly this. If scene 42 is continuous with scene 41, shot yesterday, the sleeve roll and the loose strand have to match what the script supervisor and wardrobe noted then.
What slows it down
The biggest problem is last looks becoming a second round of full hair and makeup because first team was called to set before they were ready. A good 1st AD asks the departments for a real "ready" time and calls first team accordingly.
Another is too many people. If eight people step in for last looks, the actor loses focus and the DP can't see the frame. Each department sends one person.
Missed calls are the third. A makeup artist who didn't hear last looks on the walkie gets to set as the 1st AD calls rolling, and now the company waits. On bigger stages, a PA says it out loud at the makeup station too. See rolling for what happens right after.