There are many elements of United States politics which are either handled through appointment or are restricted to a limited class of voters. First, voting is often geographically limited, e.g. if you are a resident of Montana, you cannot vote for the Senator for New York. Voting can also be limited in “closed” primaries to registered members of a specific political party. Some states have closed primaries in which, for example, only a registered Republican may vote for candidates to represent the Republican party in Presidential elections. When an elected official dies or resigns during her term of office, a temporary successor will often be appointed to serve out the term. Finally, many important positions in the judiciary are handled by appointment.
Anything further or specific with regards to example positions that are guarded from various voting classes?