Any way, written as two words, is the noun way with the quantifier any as its determiner.
I don't know any way to fix it, but George may know a way to fix it.
There isn't any way I know of from here to there.
Anyway and anyways are adverbs. At one they were equivalent to in any way = by any means, in any manner, but in Modern English they're mostly† restricted to a contrastive sense approximately equivalent to nonetheless or regardless.
[They told us not to do it, but] we're going to do it anyway.
It's often used to resume the thread of one's discourse after a digression, with a similar contrastive sense of "you can disregard all that".
Adverbial anyway no longer has any undertone of the original meaning, and it is spoken differently, with -way unstressed—which is probably why people started writing it differently.
The version with -s is a hangover from the Old and Middle English "adverbial genitive", in which the genitive form gave an adverbial sense. Some of these have survived in Standard English—always, once, hence, towards, and expressions like *he works nights—but the construction is not productive today, and most of these -s forms appear only in dialect. Anyways is one of those; it is not used in Standard speech or writing.
† You occasionally find uses in the old sense. For instance, They scrape by anyway may mean "They survive by any means they can", and "He does his work anyway* may mean "He does his work however it suits him, without any regard to its quality".