My dictionary says 助動詞{じょどうし} means "auxiliary verb". This sounds intuitively correct; 動詞 means verb, so 助動詞 sounds like a type of verb.
But the 助動詞「ない」 looks like an auxiliary adjective, not an auxiliary verb.
This is confusing. I suppose I don't know exactly what a 助動詞 is. Does it mean that ない is a type of verb? That seems strange, because it doesn't look much like a verb. Or is 助動詞 a wider category than just "auxiliary verbs"? That seems strange, too, because they're called 動詞.
What exactly is ない?