What are tannins?
A group of naturally occurring compounds — polyphenols — that bind to proteins, which is why they produce a drying, gripping sensation in the mouth rather than a taste.
Why they feel rather than taste. Tannins bind to the proteins in your saliva and to the surface of your mouth, reducing lubrication. The result is astringency — a rough, puckering, drying feeling. It is a tactile sensation, not one of the basic tastes, which is why describing tannin as "bitter" is imprecise, though many tannic compounds are also somewhat bitter.
Where they come from in wine: grape skins, seeds and stems, and additionally from oak barrels. This is the fundamental reason red wine is tannic and white generally is not — red wines are fermented in contact with the skins, extracting colour and tannin together, while white grapes are usually pressed and the juice fermented alone.
Longer skin contact, thicker-skinned varieties and a higher proportion of new oak all increase tannin.
Why tannins matter:
Structure. They give a wine grip and a framework, which is why a tannic wine feels substantial and a wine with none can taste flabby.
Ageing. Tannins are antioxidants and polymerise slowly over years, gradually softening and precipitating out as sediment. This is a primary reason some wines improve with age and others do not — a wine without tannin has little to develop, which is why most whites are drunk young.
Food pairing. Tannins bind to proteins and fats in food as well as in saliva, which is why a tannic red is softened by fatty meat and cheese, and why the same wine tastes harsh on its own. This is a genuine chemical interaction rather than a convention.
Where else you encounter them: tea, prominently — which is why over-brewed tea is astringent and why milk, being protein-rich, softens it; unripe fruit; walnuts; and dark chocolate.
How to reduce perceived tannin: decanting, which helps somewhat; serving slightly cooler, since warmth accentuates them; and eating something fatty alongside.