All riders should wear a boot with a heel. The heel acts like a stop, preventing the foot from getting caught in the stirrup. If a rider gets thrown, but the foot is still stuck in the stirrup, the rider could be dragged to death.
Boots don't have to be any particular style or color. You can wear a western or English-style boot. Manufacturers have even developed "riding sneakers," which look like high-top athletic shoes and are an affordable alternative.
Boots also protect you when you're on the ground. Horses can sometimes forget where your feet are and place a hoof on them. If you're just wearing sandals or a thin little tennis shoe, your toes could be crushed. Boots can offer your feet some protection.
The western boot of choice is a lace-up ankle boot called a roper. English riders have a lace-up ankle boot, as well, called a paddock boot. They also have a choice of tall boots that come in two different styles: field boots (lacing at the top of the foot) and dress boots (without laces).