Who selects Supreme Court nominees?

Each state sets its own policy about selection of justices for its supreme court. Some follow the US example, with the governor nominating candidates and the state legislature voting to approve or reject. Other states follow something called the "Missouri Plan," in which a special Committee presents three potential nominees to the Governor, who either chooses one or defers to the Committee selection. The justice is then appointed to a temporary one-year term, after which he or she must be reconfirmed by popular vote in a general election every X-number of years. Some states use the popular vote exclusively; in those cases, the state determines whether the candidate may be affiliated with a particular party, or whether he or she must run as a non-partisan. Some states use general election processes (both partisan and non-partisan), and there may be other variations.

The question is really too general for more specific details; for more information, please specify the state in which you're interested.