My cheek hurts really bad is this a gum or mouth infection?

Could be either +++. When cheek hurts, problem could be abnormality on the cheek itself (biting own cheek, cyst, inflammation etc), teeth on that side, referred sinus pain, abnormality of the nerves leading to the cheek. Try good oral hygiene first: brush properly with soft toothbrush and floss regularly, gently brush tongue and cheek, warm water rinses, etc. If problem persists--general doc, dentist, referral prn.