It’s easy to confuse the meaning of prone and supine — and it’s important to distinguish between them, because they’re antonyms. (I also discuss here some of the synonyms of each word.)
Prone, from the Latin term pronus, means “inclined to,” and it is commonly used in this figurative sense as well as to mean “lying face down.” Pronate, used both as a verb and as an adjective, means “to bend forward” or “bent forward,” respectively.
Prostrate, a synonym for prone, means not only “lying flat”; it has the additional connotation of “stretched out” and often refers to the adoption of that position to indicate submission, as a subject lying prostrate before a monarch. (Prostrate, not prostate; that’s the name of a gland in male mammals.) Prostrate is also the verb form, and prostration is the noun form. (Prostration is not to be confused with obeisance, which refers to the mere act of bowing.) Procumbent is another synonym; it also describes nonrooting plant stems that trail along the ground.
Supine, from the Latin word supinus, means “thrown or turned backward,” and describes someone who is lying on one’s back; unlike prone, it has no figurative sense. Supinate is also an adjectival form, and supination is a noun meaning “the act or state of lying on one’s back.”
Recumbent is a synonym that also suggests the act of leaning back or resting, as on a bed or couch; in addition, it describes such a pose in visual art. Decumbent, meanwhile, also means lying down and in botany denotes a plant that does that but has vertical parts. (Yes, incumbent, meaning “one who occupies an office or position,” is related to the other -cumbent terms here.)
Prone and supine each have rarely used adverbial and noun forms: pronely (or simply prone) and supinely, and proneness and supineness.
Pronation and supination are used in anatomical and medical contexts to refer to the position of limbs, especially, in sports medicine, to the placement of the foot while running; supination (or underpronation) can cause injury.
Supine also has a meaning as a noun; it refers to an infinitive phrase starting with to or, in Latin, to a specific type of noun.
Two terms similar to prone and supine are dorsal and ventral; dorsal refers to the back, and ventral refers to the abdomen. To help you remember which is which, think of how the first syllable of dorsal rhymes with porpoise, distinguished by its dorsal fin. Ventral, meanwhile, though its first syllable is not etymologically related to vent, can be remembered as the side from which you breathe.
Mnemonic clues to help you remember which is which include thinking of the pro- in prone (which actually means “forward”) to remind you that when you are prone, your face is toward the floor or ground. Supine, meanwhile, can be related to spine, which when you are supine is in contact with the floor or ground.