They're astringent, but ...
Tannins were used to tan hides, way back when. And the extremely low-rent part of town was right next to the tanning district ... think enormous amounts of rotting pee with rotting hides and oak bark in lots of open vats, and you get some idea of what the smell of said tanning district must have been like.
The oak bark in above example supplied the tannins, but I'm sure that other parts of other plants were used as well.
There's gallic acid, and there's tannic acid, and then there's the funny fact that quercetin is the strongest flavonoid there is -- or at least, it used to be. It's entirely possible that they've found a stronger one by now.
And quercetin is found in one of the strongest tanning plant parts there is - oak bark. The name quercetin tells you where it's from: Quercus is oak.
So, as far as I've been able to find out, some tannins are concentrated flavonoids. I haven't managed to find out if any of the flavonoid action of quercetin makes it beyond the enormously strong astringency of oak bark, though.
Tannins are astringents. That's pretty much it, for most of them. You can read all about astringent action in some of my older blogposts, for instance those mentioned in the "other entries", below.
Finally, I know that this particular blogpost is as flimsy as they get. So shoot me: I'm a herbalist, not a pharmacognosist.