Who can vote in a Primary?

Answer 1

Depends on the state.

It can go from: A registered voter walks into a polling place and says I want to vote (and they can for anybody in any office, regardless of party affiliation).

to: You have to register in a party [at least 30 to 60 days in advance] and can only vote in that party's primary.

Answer 2

Depending on the state, all registered voters, or more commonly, all registered voters of an individual party. So if you are a Democrat, you vote in the Democratic primary.