There are three answers to this question: a heck of a lot, not that many, and a whole heck of a lot. Or, if you want specifics: 5, 2, and 99. Confused? The question has been problematic, and the best way to understand what the answers mean is to take a look at the history of people talking about Eskimo words for snow.
There is no single "Eskimo" language. "Eskimo" is a loose term for the Inuit and Yupik peoples living in the polar regions of Alaska, Canada, Greenland, and Siberia. They speak a variety of languages, the larger ones being Central Alaskan Yup'ik, West Greenlandic (Kalaallisut), and Inuktitut. There are multiple dialects of each. Some have more words for snow than others.
A heck of a lot
Today, you see the "Eskimos have so many words for snow" trope everywhere from ads to cartoons to articles about hairstyles. As Laura Martin noted in her 1986 article "Eskimo Words for Snow," anthropologists and psychologists started using the story in the late 1950's as a go-to illustration in discussions of the relationship between language, culture, and perception. If Eskimos carved up the world of snow into four or five categories where we had one, was their perception of snow different from ours? From there the idea spread into the popular culture, and it has been going strong ever since. Where the original sources mentioned four or five specific snow words, in the hands of the general public that number turned into 25, 50, 100, 400 – it didn't really matter. The story did not exist to give information about Eskimo languages, but to say, "hey, other people sure do look at the world differently!"
And this was problematic. The idea of using language to show that other people look at the world differently had a nasty history. Early ethnographers used linguistic evidence to impugn the character or cognitive abilities other peoples. An 1827 book mentions that in the language of Lapland "there are five words for snow, seven or eight for a mountain, but honesty, virtue and conscience must be expressed by a periphrasis." The academics who picked up the snow words tale in the 1950s didn't take such a simplistic view of the relationship between language and culture. But to say that having a lot words for something means you find it important or perceive it more readily, gives some people the wrong idea that that not having a lot of words for something means you can't perceive it and don't find it important.
Not that many
Part of the debunking of that false implication came in the form of a debunking of the snow words trope. Martin's paper and Geoffrey Pullum's well-known essay "The Great Eskimo Vocabulary Hoax," pointed out that the linguistic facts did not support the idea that Eskimos had some wildly exotic giant snow vocabulary.
The Inuit and Yupik languages are polysynthetic. Polysynthetic languages combine a limited set of roots and word endings to create an unlimited set of words. For instance from oqaq – the West Greenlandic root for "tongue" – you get oqaaseq (word), oqaasipiluuppaa (harangues him), oqaluppoq (speaks), oqaatiginerluppaa (speaks badly about him) and Oqaasileriffik (Greenlandic language secretariat). These can then be expanded with all sorts of other endings, so that a sentence like, "I hadn't planned to cause you to harangue him after all" would be expressed with one word. If these word-sentences count as words, then Eskimos don't just have thousands of words for snow, but for everything.
Martin suggests that we instead ask how many roots Eskimos have for snow. In the case of West Greenlandic, the answer is two: qanik (snow in the air), and aput (snow on the ground). From these we can get derived words like qanipalaat (feathery clumps of falling snow) and apusiniq (snowdrift). There are also terms for snow that use different roots (for "covering," "floating" or other things snow does), but Pullum's essay notes a problem with the notion of counting words with other roots as "snow words": Do we count an Inuit word that can mean "snow for igloo making" as a snow word if it also just means building materials in general? To use another example, is "pack" a snow word in English, or a just a general term for tightly smushing things? In any case, there may be just as many snow words in English (sleet, slush, flurry, avalanche, etc.) as in "Eskimo" languages.
A whole heck of a lot
The linguist K. David Harrison has traveled all over the world studying endangered languages. In his book The Last Speakers, he says it's a mistake to think that just because people made uninformed and exaggerated claims about Eskimo snow words in the past, the real number must be ordinary and uninteresting.
From what he has seen, "the number of snow/ice/wind/weather terms in some Arctic languages is impressively vast, rich, and complex." The Yupik, for example, "identify and name at least 99 distinct sea ice formations." For example, there is a word Nuyileq, meaning "crushed ice beginning to spread out; dangerous to walk on. The ice is dissolving, but still has not dispersed in water, although it is vulnerable for one to fall through and to sink. Sometimes seals can even surface on this ice because the water is starting to appear."
Clearly, there is a lot more included in this definition than would be included in a typical dictionary definition. But it shows how a set of terminology can reflect a complicated body of specific expertise. Every area of expertise has such a set. Geologists have lots of words for rocks, linguists have lots of words for speech sounds. This means Eskimos may not be any more exotic than geologists or linguists, but it does not mean their words for snow are uninteresting. You can learn a lot about what distinctions are important to make in a field by looking at the distinctions between words. The Yupik ice words, whatever the number, are important because they package information in a useful way. We ignore the significance of that packaging, as Harrison says, "at our peril."