It depends on the court and the state. Judges are expected to decide cases impartially, but in the United States some judges are still chosen through elections in which their party affiliation appears on the ballot. Federal judges, including U.S. Supreme Court justices, are not elected at all: they are nominated by the President and confirmed by the Senate.
State courts vary widely. According to the Brennan Center for Justice, 39 states use some form of election for at least some judges. For state high courts, 14 states hold contested nonpartisan elections (no party label on the ballot), 8 states hold contested partisan elections, and 16 states appoint justices and then keep them through unopposed yes-or-no retention elections. Partisan elections are more common for trial courts, which 11 states fill that way, while 21 states use nonpartisan elections for trial judges.
Source: Significant Figures in Judicial Selection, Brennan Center for Justice
Image: Courtroom One Gavel by Joe Gratz, CC0, via Wikimedia Commons.