Note that modern grammarians prefer to distinguish the class of a word from its function in any particular context. (Actually, there are some controversial theoretical issues about what it really means for a word to belong to a certain class, but I don't understand these well enough to comment about them, and I don't think your question requires getting into that topic.) Classes of words can be nested, and there are many many classes: far more than the familiar eight or so that most people are taught about in English classes. Unfortunately, the names given to particular classes are not always uniform between authors.
The word "three" belongs to the class of numerals. In a comment, BillJ says that the class of numerals in turn belong to the class of determinatives, which seems to be a fairly standard analysis. Numerals are also sometimes said to belong to the class of quantifiers, a subset of determiners (SIL Glossary of Linguistic Terms), although confusingly the Wikipedia article on quantifiers suggests that numerals are excluded; I don't know if that bit was put in by someone who knows what they're talking about or if it's just the result of someone being confused.
Determinatives themselves can be considered to belong to an even larger overarching class of words that includes things like common nouns and adjectives, but as far as I know most homework about identifying the class of words would want you to give answers of greater specificity than that (also, there doesn't seem to be any unambiguous name for this overarching class that is in common use; "An Introduction to Functional Grammar" by Halliday and Matthiessen (2004) says that numerals, common nouns, adjectives, and determiners are all subclasses of a primary class that Halliday and Matthiessen label noun, although they say that it is sometimes called "nominal" to avoid confusion with the narrow sense of the word "noun" that is understood to exclude things like adjectives).
I'm not sure how to categorize the function of "three" in the phrase "his three big dogs", but I guess technically, your question doesn't ask about that.