When one thinks of a cart wheel, one is thinking of a wheel for an old-fashioned cart for a horse. The wheels are quite big, so as to lift the cart far enough off the ground to 1) tether comfortably to the horse, 2) avoid rocks in the road, and 3) still retain height for the cart when the wheels go down into the ruts in the road. The wheel is made of hard wood or metal, with a hub in the middle and a hole for the axle to stick through, a big circle on the outside, and long spokes to connect the hub and circle together. The big circle turns around on the ground as the cart is pulled forward.
A cartwheel is the name for a gymnastics movement which looks like a circular cart wheel going around. You start in a standing position, turn over in a circular movement sideways with your arms and legs out wide like cartwheel spokes, and end in a standing position again. A person well practised in cartwheels can continue going over and over, turning multiple cartwheels in a row and ending up a fair distance to the side of where they started.
The movement for a cartwheel should be fairly quick as it requires momentum to achieve succesfully. Say you are going to do a cartwheel to the left. From a standing position, take a step to the left side with your left leg, start to lift your right leg up sideways to the right, put your left hand down on the left side with your fingers pointing towards the back. Use the momentum created to also lift your left leg up in the air as your right hand is placed on the ground to the other side of your left hand. Your legs continue up and over; in the middle of the movement, it will look like a snapshot of a handstand with your legs straddled wide. Then as your leading leg starts to come down on the other side, your first hand (left) lifts up, your first leg (left) touches the ground, your second hand (right) lifts up and your second leg (right) touches the ground. Ta da! One rotation of a cartwheel performed successfully.
A fellow noder has sent me a message, questioning the above instructions. There is apparently another way to perform a cartwheel, and that is to start forwards and end facing backwards. I learnt the side way, and did a bunch of them while writing the above to make sure I described it properly. The advantage of the side way is that the flow of multiple cartwheels is better and more continuous. The advantage of the forward way is a cleaner movement and less chance of falling over.