You either register with the Post Office(USPS) or with your city's/county's/state's government. For California you can see the registration form and options here(just click register and say yes and you'll see the form with options): https://registertovote.ca.gov/

You register to vote and, depending on state/county, you choose what party you belong to(Republican, Democrat, No Party, Green Party, etc.). You may need to register with a specific political party because of open vs. closed primaries where you MUST be a registered Republican to vote in the Republican primary and/or register Democrat to vote in the Democrat primary, etc.

California has a good website/page with more details: https://www.sos.ca.gov/elections/political-parties/no-party-preference/

Again, different states have different ways of doing things.