Cutting is competitive herding of cattle. It's so named because the horse will "cut" one particular cow from the herd.
It might seem strange to use one herbivore to herd another, but "cow sense" - or herding instinct - has been bred into the stock horse breeds over generations. The original instinct probably came from a dominant animal's desire to keep their own herd together. (Horses with a lot of "cow" also tend to be more dominant personalities, which they need to face down a steer - but will often try to use to face down their handler).
In a cutting contest, the rider will generally cut three selected cows from the herd. The horse (with no guidance from the rider) will then try to keep the cow separated. The original purpose of cutting was to separate cows for doctoring (vaccinations, etc).