What it is
Wine begins as nothing more than fermented grape juice, so the vegan question is not really about what is in the bottle but about how it was cleared. Young wine is often cloudy with proteins, tannins and spent yeast, and to polish it bright a winemaker adds a fining agent that latches onto those particles and drags them to the bottom, where they can be racked or filtered away. The classic fining agents are animal: gelatine, isinglass (a collagen from fish swim bladders), casein from milk, and egg white, each picked for the particular haze or harshness it removes. Their defenders note, fairly, that the agent is mostly pulled back out again before bottling, so at most a trace could remain; but for most vegans an animal product used in the making is enough to set a wine aside. The animal-free options do the same job perfectly well, chiefly bentonite, a clay that draws proteins out of white wine by electrical charge, along with activated carbon, plant proteins, and simply giving the wine time to settle on its own.
Why the label won't tell you
Here is the frustrating bit: because fining agents count as processing aids that get filtered back out, most countries do not require them to be named, so a bottle fined with isinglass can still look, from its label, like pure grape juice. The EU and Australia are partial exceptions, but only for allergens, which means a wine fined with egg or milk protein may carry an 'egg' or 'milk' warning while fish isinglass and pork gelatine pass unmentioned. That leaves a few reliable ways to check. More and more producers now print 'vegan' or 'suitable for vegans' on the back label, and 'unfined' usually points the same way; failing that, free directories and apps that catalogue drinks by their fining practice, built from information wineries supply, will often have the answer. When in doubt, the winery's own enquiries line is the last word, because the person who clarified the wine knows exactly what went into it.