The taste buds may detect sweet, salty, sour and bitter, but what goes on in our mouths when we experience spicy or minty tastes?

The pain receptors on our tongues, not the taste buds, are stimulated by the chemical capsaicin, which is in spicy food, and sends messages to the brain. Capsaicin is the chemical in capsicum spray by the way, and our eyes are also very sensitive to it. Menthol in mint also stimulates pain receptors.

Just to add that Japanese identify a fifth type of taste 'Umami', imparted due to the presence of glutamic acid and ribonucleotide guanylate in some mushroom and meat stuffs
http://www.mushroomsonthemenu.com/umami/

Elaborating on what onlytrivial has already said...

" A chemical sensor, working alongside smell and taste, is the trigeminal sense. Trigeminal (cranial V) nerve endings in the tongue and oral cavity sense, e.g., pungent chemicals given off by such "hot" spices as red chili pepper (capsaicin), black pepper (piperine), mustards and horseradish (isothiocyanates), and onions (diallyl sulfide). They also respond to "cool" spices, such as mint (menthol), and to the chemical "bite" or "sharpness" of ethyl alcohol in tequila and rum. In each of these cases, our trigeminal nerve endings respond to chemical irritants rather than to gustatory taste cues per se (which are sensed instead by the facial nerve [cranial VII])."

http://www.hotstream.org/9961/why-does-spicy-taste-hot-and-minty-taste-cool/