When it is available for booking, you can book online or by phone.
It is not held every night. If you call, the CMs can often look ahead in their calendars and tell you whether it will be offered on a specific night even when that night is not yet available for booking. They *seem* to be releasing booking dates *about* 90 days in advance but that is by no means certain. And if you call and ask, you'll probably just get the wrong answer. Best advice is to check online or on the Dis fairly often; when new dates/months open up, somebody usually starts a thread here.
The Dessert Party is not part of any of the dining plans and they do charge your credit card in full when you make the reservation. It is a real reservation, not an ADR. No matter what time they give you when you book it, the party starts for everyone 1 hour before that night's fireworks. You have an actual table reserved that has you name on it when you go in.