The key is not to confuse part of speech with function (I just gave a lecture on this toady). The "word" is a conjunction. There are three basic types of conjunctions: coordinating (and, but, for, not, etc...), suboordinating (after, although, as, because, before, how, if, once, since, than, that, though, until, when, where, whether, and while), and correlative (both...and, either...or, neither...nor, not only...but also, so...as, and whether...or).

Coordinating conjunctions are used to join individual words, phrases, and independent clauses. For example, I didn't wake up on time SO i was late.

A subordinating conjunction introduces a dependent clause and indicates the nature of the relationship among the independent clause(s) and the dependent clause(s). For example, “Begin grooming your kitten while it is still young.”

Correlative conjunctions always appear in pairs -- you use them to link equivalent sentence elements. For example, "I am neither happy nor saddened by the news about the change in the school calendar."

"While" is a conjunction. Specifically, it is a coordinating conjunction that joins the independent clause "Begin grooming your kitten" and "it is still young." By adding "while," it forms a dependent or suboordinating clause.

It is true that the function of the clause is adverbial, but that doesn't change the part of speech of the word. There are three types of dependent or suboordinate clauses: 1) noun, 2) adjective, and 3) adverb. Since the whole clause performs the function of an adverb (indicating a time that a predicate should be performed), the whole "clause" is adverbial; however, the word "while" is still a conjunction within that clause.

I hope this helps!