General elections are public elections in which anybody who meets certain requirements and is willing to invest the effort can run. As a public election any voter can vote for any candidate. If the candidate has been nominated by a political party, that party will be indicated on the ballot (and most Americans simply vote or the candidate nominated by their party of choice).
Primary elections are basically a private affair undertaken by a political party in order to choose a candidate who they will nominate for a general election. Primaries are facilitated by the public voting system and tend to be influenced by lots of state laws, but most of the specific rules governing a primary are left to the political party (and sometimes delegated to that party's state level organizations).
So parties get to make all kinds of rules about their primaries. In many states (it is subject to state law) parties limit participation in their primary to their members, or more often they disallow members of other parties (but allow someone who hasn't declared a party affiliation). The theory seems to be that this makes it more difficult for one party to ask their members to vote in another party's primary, in an attempt to nominate a candidate who wouldn't be viable in the general election.