Roping

There were three types of roping that I saw at the rodeo: Calf Roping/tie down, breakaway roping, and team roping.
In calf roping/tie down roping, the calf started in the chute, with a light rope(string like) wrapped around its neck.

Once the rider nodded his head, the chute opened, and the calf started to run. Once the calf got to the end of the rope, the barrier in front of the horse came down, and the horse flew after the calf. The rider threw the rope and if it landed on the calf, the horse stopped, while the rider dismounted and threw the calf, tying it.

For a better explanation of the event there's a great post on Braymere Custom Saddlery
Breakaway roping was similar to the calf roping/tie down roping, except for the rider roped the calf, pulled the horse to a stop, while the calf continued forward until the rope 'broke' and released the calf to continue running.

Team roping consisted of two ropers.

The 'header' or first roper, roped either both horns, one horn and the head, or the neck. 

The header turned the steer, and the 'heeler' or second roper, roped the steer's hind legs.