Generally, elected.
If there is an opening before the end of that Justice's eight year term, the Governor can appoint a replacement for that slot. However, the appointee must run in the next election to fulfill the remainder of the eight year term.
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In the U.S., it varies by state. Federal judges are not elected; they are appointed.
Depending on the state, judges may be elected, appointed, nominated and confirmed by the legislator or nominated by the state bar.
it provided the framework for popular sovereignty
The only branch directly elected by the people is the house of Representatives. Judges are appointed and the president is elected through electoral college