When I first encountered the idea of vegan-friendly beer my reaction was something along the lines of WTF!?! Beer is water, grains, hops and yeast, right? It's not like there are big chunks of juicy steak floating in it!
Well, as it often is, the answer is more complex than that. The truth of the matter lies in two things, adjuncts and finings.
First, let's define vegan. I'm going with vegan because, at least in my mind, it is the easiest form of vegetarianism to deal with.
When you start looking into the vegetarian lifestyle, you find that there many different levels, if you will. But, parsing degrees of vegetarianism isn't our interest here. So, rather than getting into the shades of gray that vegetarians deal with, we're going with a straight up vegan which allows no products or byproducts that are of animal origin.
Now, adjuncts produce some pretty obvious non-vegan beers. Honey beers, for example, are brewed with honey and are naturally excluded. It isn't always that obvious, though. Sometimes beers will be named after things that they are associated with but do not include. Cream and oyster can appear right there on the label and yet no cow labor or oyster fatalities were involved in the brewing of that particular beer. On the other hand, there are many examples of milk and oyster stouts that do include these ingredients so it's up to you to investigate.
Fortunately, the good beer movement makes your investigation as easy as turning the bottle over and reading the label in many cases. Smart brewers know that they are dealing with a more specialized market these days and want to provide as much information as they can to the customer standing in the beer store.
Things aren't quite so clear on the finings side. Traditional finings, which is used to clarify beer, include gelatin and isinglass. Since both of these are animal byproducts the resulting beer is not vegan-friendly. Fortunately for beer loving vegans, these are mostly used by the very traditional brewers. Modern filtration equipment makes the use of less efficient but easier to store adjuncts possible with equal or even better results. In my experience, most of the today's brewers use the later method of fining and filtration.
The investigation on this end of the vegan beer question isn't quite so easy. Since finings falls out of beer and doesn't make it into the final product, they aren't considered ingredients in the traditional sense. But, by vegan standards, they are so, how to avoid them? There is a website, barnivore.com, which tries to keep up with which breweries brew vegan-friendly beer. The brewing community is so fluid, though, that this seems to be a nearly impossible task. While in most cases you can probably rely on carnivore's list, the only way to be absolutely sure is to call the brewery and ask.
Check out The Spruce's guide to vegetarianism for more about vegan beer (and vegan wine).
Also, there is an annual vegan beer festival held in California every summer. Check out Los Angeles Vegan Beer Fest for more information.