Tannins are found in the skins, pips and stems of grapes. They add a texture in your mouth which is like the drying effect on your gums of stewed tea.
In white wine making, the aim is to separate the juice of the grapes quickly from the pips and stems to avoid any tannic or woody sensations.
In red wine making, on the other hand, winemakers like to play with the amount of contact the grape juice has with the skins and pips. If the grapes are ripe, and the pips have turned brown, then the tannic flavour and texture can add character to a wine. It is also one of the elements that is needed, along with a balance of fruit and acidity, in a wine that is intended for long cellaring.
There are a number of terms used to describe tannins. 'Soft, supple, velvety' describe a smooth texture in the mouth. This is a sensation that can be found after several years of ageing as the harshness of the young wine softens. But it can also be found in young wines, depending on how the winemaker handles the grapes.
'Firm' tannins are also said to have 'grip'. This means that you can feel the drying sensation on your gums. In this case the wine is probably young and will be best enjoyed with food which can soften the harsh feeling. This sense of grip in the mouth is welcome with big, bold reds; it often makes a good balance with rich fruit and high alcohol, and will soften with time.