"The Earth spins on it's axis" is the most common. "The Earth rotates on it's axis" is less common but it can also be used. Both SPIN and ROTATE give the impression of an object turning all by itself. The phrase "spin the dial on the machine" for example. "The Earth revolves around the sun" is very common. "The Earth circles the sun" can also be used. REVOLVE and CIRCLE give the impression of an object travelling around some other object which is separate from itself. "I wanted to avoid the traffic, so I circled the city" means drove around the city. So the Earth spins on it's axis as it revolves around the Sun. The tires rotate on my bike as I circle the park.

These words are close enough in meaning that a person should know what you mean in any case. However, sometimes one word is more commonly used than the others. "Spin" is most commonly used when the object is turning around very quickly, but moving little from one location, especially in a way that stabilizes an object. For example, a coin which is "spinning" will be able to stand on its edge for a time. "Rotate" is more common when the object is spinning slowly. Also, scientists should always use "rotate" rather than "revolve" when talking about something in outer space which is slowly spinning. For example, "The earth rotates on its axis." Revolve is much more common when an object moves around another object or position. It leaves one location, travels in a circle, and then back to its original location. This should be used by scientists when talking about an object that orbits another object. For example, "The earth revolves around the sun." "Circle" never means "spin". It almost always means to move location around another object or position. It can also mean to draw or create the shape of a circle out of something. For example, when using a pen you might "circle" something on a paper, meaning you draw a circle around that thing.