Tannins are natural compounds found in grape skins, seeds and stems, and in the oak barrels some wines are aged in. They’re what makes a wine feel dry and puckery, the same grippy feeling you get from a strong cup of black tea.
You feel tannins more than you taste them. They bind with proteins in your saliva, which is what creates that dry, astringent sensation.
Red wines have far more tannin than whites, because red wine is made by fermenting the juice with the skins. White wine grapes are usually pressed right away and separated from the skins. But some whites do pick up tannin, from oak barrels or from time on the skins, so it isn’t strictly “only found in red wines.”
Why they matter: tannins give a wine structure and help it age. As a wine matures, the tannins gradually soften, so older reds feel smoother.
Tannin vs. acidity: they’re easy to mix up. Acidity makes your mouth water and tastes tart and fresh. Tannin makes your mouth feel dry and grippy.
With cheese: tannic red wines do well with rich, fatty foods, which is one reason a bold red is a good match for an aged hard cheese. See What wine or whiskey goes with cheese?