It depends. Some cities do nonpartisan elections (usually with some form of "jungle primary"), some cities do partisan ones. For the ones with partisan elections, there are often more than two parties, but parties other than the Republicans and Democrats rarely have a shot at office.
NY has a unique system in that it has a lot of parties, but many of them just endorse a candidate from one of the big 2 parties rather than running their own. This is why Cuomo isn't entirely out of this thing yet, unfortunately; he conceded the Democratic nomination, but he still has the nomination from the Fight and Deliver party (which he, himself, founded specifically for this election) and may choose to continue to run as their candidate instead of dropping out.
Apparently they upheld the decision on appeal, which is just fucking wild.
Does YouTube have bots handling the appeals or something?