Believe it or not, not all wine is vegan — or even vegetarian. The products often used to fine and filter wine (the things that give it that crystal-clear color) are made from animal ingredients.
If you ever see wine that's murky or cloudy, it's because the wine wasn’t filtered. The cloudiness can come from yeast, proteins, or organic particles. And although these things won't hurt you, most people prefer a clear wine.
In order to make your wine clear, winemakers put it through a fining process. The most popular fining techniques involve animal-derived products made from egg whites, gelatin, fish bladder, or casein. Although these products do come out of the wine along with all the small particles that stick to them, it's possible for traces of the fining agents to be absorbed into the wine — hence not all wine being vegan.
Vegan wines usually still go through the fining process, but winemakers will either use mineral products or manually filter the wines to make them clear.
Also, just because a wine is organic doesn't mean it's vegan. Some organic wines are vegan, but not all. However, as more and more winemakers have begun crafting organic wines, the industry's also seen a shift toward more vegan wines.
It can sometimes be difficult to tell if a wine is vegan or not, because not all vegan wines are marked as such. There are some online resources such as the Vegan Wine Guide and Barnivore — or you can contact the winery directly to find out.