It's thought that there are some planets like this. They are called "orphan planets" or "rogue planets" and drift aimlessly through space. They are not near any stars and therefore do not receive the large amounts of radiation that we get from the sun. No life like our own could exist on the surface, since we require energy from the sun as the base of our food chains. It is thought there could be heat-loving microbes or life forms on these planets since they would still have tectonic plate motion. Some scientists also have shown that they could have heavy atmospheres that trap in geological heat, allowing the planet to even be warm. They can form by being flung out of a developing solar system. Since we haven't yet identified one, we don't know many details about what these planets could be like.

Without the sun, the Earth would freeze into a ball of ice. Without the moon, the Earth would have much weaker tides, and those tides would be from the sun. Eventually, the Earth's rotation would slow down until one side of the Earth always faced the sun while the other side always faced away from it. Instead, we are slowly locking onto the moon, instead of the sun.

This is a complex question to answer since so much would depend on how the sun or the moon disappeared. If our sun never formed, than the Earth would not have had a gravity well to rotate around and would likely never have formed. If our moon never formed, we’d still have life on Earth but it would be quite different from what it looks like today. And, if either disappeared, the results would be catastrophic. For in-depth discussion on what would happen if they did, check out these sites:

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