Chute dogging is an event where a you will guide a steer out of a chute and wrestle it to the ground. It's similar to steer wrestling, but you start in the chute with the steer instead of on horseback. Typically this event is for boys. 

The object is to race to a goat staked out on a  rope in the arena, flank the goat, then gather 3 of its legs and tie them together. Typically this event is for girls. 

Horse riding with a lead rope (a long rope attached to a horse's halter used by a handler who will guide you.) This event is for both boys and girls. 

Roping a calf dummy, also known as practicing with a non-live calf, is a valuable tool to learn the skill without the horse or calf. This event is for both boys and girls.