Yes, most states request that you indicate your party affiliation in order to vote in primary elections which are often organized by party. (This is not the case for general elections where party affiliation is not required.)
Not all states hold primary elections by party, but most states who do also ask that you indicate the name of your political party, or the name of a non-qualified political body into the Voter Registration/Absentee Ballot Request form. For example: Democratic, Republican, American Independent, Green, Libertarian, Natural Law or Peace and Freedom, etc.
If you have not chosen a political party or do not wish to declare your party, indicate that with an appropriate note such as "decline to state," or "none, or "nonpartisan", or "undeclared."
Most Secretary of State websites provide further information about which political parties are allowing nonaffiliated voters to participate in their primary election.