It depends on the city. Mine (Madison, WI) doesn’t officially list any party affiliation for mayor or city alders. We also use a runoff election system, so we’re not stuck on two parties for local things.

In practice, candidates are often backed and/or endorsed by some political parties. Common ones are Progressive Dane (county level party) or Working Families (which has national reach and is basically a socialist party working within the Democratic party). When they move up to state or federal seats, they usually join the Democrats while continuing to work with the Working Families party.