A ball. Soccer balls come in different sizes depending on how old you are. Kids 8-12 should use a size 4 ball, and kids 13 and over should use a size 5 ball. Synthetic leather balls are best for beginners, because they don't absorb water and get heavy.
If you play in a league, a goal will usually be provided for you, and you can buy a smaller goal if you want to play in your backyard — just make sure it is anchored to the ground. No goal? No problem! Just set up any two objects (cones or waterbottles are good) to shoot between.
Two pieces of equipment you need to wear at all times when playing soccer are shin guards and cleats. Shin guards are designed to protect your legs from the ball, and from being kicked by other players. They are required in most leagues. The right cleats to wear for soccer are ones that are plastic or rubber — they'll help you with your quick starts, stops and turns.