Tannins are phenols that occur naturally in grape skins and stems. In wine, tannins come from leaving the skins and stems in contact with the juice after pressing. Tannins are what makes your mouth feel dry when tasting a full-bodied red wine (or drinking a cup of tea). Harsher tannins often also cause your cheeks to pucker a bit, or you may feel them in your jaw, similar to when you eat something slightly sour.
The tannins in wine create structure; that is, they allow the wine to age and improve in quality as it does. Young, tannic wines are often difficult to drink because the tannins overpower the wine's subtler flavors. However, as these wines age, the tannins soften and many flavors emerge from behind them to create a nuanced taste experience.