Running your pool filter primarily involves setting it to the correct position for filtering, which is typically the most frequently used setting. This position is used both for regular filtration and when you're vacuuming the pool.
Essentially, to "run" the pool filter means to ensure it is in the "filter" position. In this mode, the pump pulls water from the pool, sends it through the filter to remove debris, and then returns the clean water to the pool. This cycle continues to keep the pool water clean. Backwashing is another common operation for pool filters, but it is not part of normal operation, and it is part of the cleaning process of the filter itself.