No reservations, first come first served. If the garage is full, there is overflow parking at the "Whale" lot. It will cost the same but not be indoor. So far, we've always been able to park in the garage, but sometimes had to look for a place.
I've also seen people parked "illegally", so I guess some don't even bother looking.
When entering the garage you pay first (credit cards okay), then drive up the ramp where porters take any checked luggage, then you park and take an elevator to check-in.