It’s the astringent, sometimes bitter, feeling of wine when every bit of moisture is sucked out. Air and aging can changes this profile greatly, which is why tannins are considered by many to be a good attribute to have in a bold red. As a big, tannic red ages, the tannins will combine with proteins in the pigment and drop to the bottom of the bottle. This softens those flavors while other fruits and tastes become more prominent. In the end, a heavily tannic wine should morph into a complex, structure filled beverage if given a few years to lay down.
Tannins can be found in white wines, but they are usually far less pronounced since the wine has little to no contact with the skins, seeds and stems in production and are usually less oaked.