What it is
Nutritional yeast is Saccharomyces cerevisiae, the same species that raises bread and ferments beer, grown deliberately on a sugar-rich medium such as molasses, then heated to kill it, washed, dried, and sold as yellow flakes. The heat step is what separates it from baking yeast: nothing in the packet is alive, so it will not raise a dough or ferment anything. What survives the drying is the cell wall and the protein inside it, which is where the savoury, almost cheesy flavour comes from.
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Nutritional yeast is a whole-cell inactive yeast that contains both soluble and insoluble parts, which is different from yeast extract.
Why yeast is not an animal
The hesitation usually comes from the word living rather than the word animal. Yeast is a single-celled fungus, and fungi sit in their own kingdom, closer to mushrooms than to anything with a nervous system. Veganism as it is normally defined avoids animal products, and no mainstream vegan organisation treats yeast, mushrooms, or bacterial cultures as animal-derived. A person who avoids yeast is following a different rule, usually a health or candida-related one, not a vegan one.
The B12 on the label
Most nooch sold as a nutritional product is fortified, meaning vitamins are added after the yeast is grown rather than produced by it. That distinction matters because B12 is the one nutrient a plant-based diet cannot reliably supply, and unfortified nutritional yeast does not contain meaningful amounts of it.
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It is often fortified with vitamin B12.
The added B12 is cyanocobalamin produced by bacterial fermentation, the same synthetic source used in supplements, so it is vegan. Check the label for the fortified line if you are counting on nooch for B12, because the unfortified version sits on the same shelf and looks identical.