Lots of clubs offer futurities for their members and all work basically the same way with some minor variations. I nominate all of my mares in the AMHR futurity - I do not have any nominated for AMHA's futurity, so will address only AMHR.
Before March 1 - you nominate your stallions at a cost of $25 each. Each mare can be nominated at a cost of $5.
This allows you to show the resulting baby at the AMHR National Futurity which is held in September in Tulsa. Pay-outs are based on the number of stallions and mares nominated and the number that actually show. In smaller classes they pay out to 6th place but in the really large classes I believe they are paying out to 10th. Some of the pay-outs were pretty substantial....there are often over 50 weanlings in a class!
You can sustain a horse in the futurity each year which then allows you to show the horse in the Yearling, 2 year old and 3 year old futurities. The sustaining fees are again EXTREMELY reasonable. $10 for a yearling, $10 for a 2 year old, and $20 for a 3 year old. 3 year olds can also compete in halter, pleasure driving, and roadster driving (correct me if I'm wrong - but only one type of driving). Each year a percentage of the nominating / sustaining fees are held back and move up with that age group so that the pay outs can be pretty large in those 3 year old futurity classes, even though the number of horses competing is fewer.
Colts can also be nominated for the Super G (gelding incentive program). They do not show until they are 3 years old and must have been gelded by that time.