When I woke up this morning under my eyes was extremely puffy and swollen. It's been stuck like that for hours! What's happening?

Many possibilities. Possibilities range from allergies to infection to any other cause of localized swelling. Depending on whether it continues or resolves on its own will determine how you need to be seeing your doctor or not. Either way if this becomes a recurrent or persistent problem, i would bring it to your doctor's attention.
? Allergic reaction. Your description sounds like an allergic reaction. Try cool compresses, chilled artificial tears, and topical and/or oral allergy meds. If no improvement, see an eye doctor.