Tannins, in the most scientific terms, are a group of bitter and astringent compounds that can be found a lot of things we like: coffee, tea, and most importantly wine.
Tannins are responsible for the actually sort of pleasant bitterness that you find in tea, coffee, dark chocolate and, of course, wine. Tannins come from four primary sources in wine: the grape skins, seeds and stems, as well as any wood barrels used during aging. They are responsible for much of the mouthfeel and weight of what you think of as "big" red wines.
While white wine is made mostly from the juice that’s pressed as soon as the grapes get to the winery, red wine is made from the entire grape. As red wine ferments, the skins, seeds, juice and sometimes stems are all macerated together. During that process, both color and tannin are leached into the wine. When a red wine is notably tannic, it’s often described as “grippy.”
Tannins are also what stain your teeth when you drink lots of red wine or coffee. Pro-tip: to avoid the coffee teeth-stain, add some full fat milk to your mug. The proteins in the milk will bond with the tannin compounds, so you will swallow and digest the compounds instead of them sticking to your teeth. But don't put milk in your red wine. That's disgusting.