There are SEVERAL types of filters, the most common being a hang on back filter(HOB), which sucks water trough mechanical, chemical and biological filter media then back into your tank. A "pump" can simply be an air pump or a water pump. You CAN have a filter running off an air pump but they are no where near as strong as propeller pumps. The most common use of an air pump though, is to power an air stone which adds oxygen to the water. A Water pump is used to push water from an under tank sump/filter back up to the aquarium.
A filter "filters" water, and a pump "pumps" (either air or water; you didn't specify)