Learn Russian: Каждый человек решает по-своему.

It is an adverb meaning "in one's own way." It is built from the prefix по- and the dative case of the reflexive possessive pronoun свой (which becomes своему). A major rule to remember is that adverbs formed with по- plus a pronoun or adjective ending in -ому or -ему are always written with a hyphen.

Whose way does по-своему refer to?

It refers back to the subject of the sentence, Каждый человек (Every person). The root comes from свой, which is a special reflexive word in Russian that means "one's own." It always belongs to whoever is doing the action in the sentence.

Could I say Все люди instead of Каждый человек here?

Yes, you could say Все люди решают по-своему (All people decide in their own way). However, Каждый specifically emphasizes "each individual" person, which highlights the uniqueness of their decisions better than grouping them all together with Все (all). Notice that каждый acts like an adjective and must agree with the singular noun человек.

Why is the imperfective verb решает used here instead of the perfective решит?

The sentence describes a general truth or a continuous, habitual fact about human nature, rather than a one-time, completed action. In Russian, the imperfective aspect is always used to state general facts and habits like this.