How do I know if a 15 year old has mouth cancer?

Proper diagnosis. Cancer of the oral cavity is rare in a teenager. However, it is certainly possible to develop leukemia or lymphoma of the tonsils. Like every disease of the mouth, which includes the lips, the cheek mucosa, the tongue and the palatine tonsils (tonsils in the back of your throat), a proper diagnosis by an ENT physician or better yet an oral surgeon is the most appropriate avenue to follow.
Unlikely. It would be rare at that age. Diagnosis requires an expert such as ENT or oral surgeon and often needs biopsy as appearnces can mislead. Tumors grow or ulcerate and do not go away over time. Have it checked as even non cancerous lesions may be a sign of other medical conditions.
Unlikely but. Unlikely See oral surgeon or ENT doc if you have any of these symptoms: Persistent mouth sore, persistent mouth pain, a lump or thickening in the cheek, a white/red patch on the gums, tongue, tonsil, or lining of the mouth, a sore throat, hoarseness or feeling that something is caught in the throat that does not go away, difficulty swallowing/chewing.