Major political parties (Democrats and Republicans) in the United States are organized from the grassroots up. The lowest level party officer is alternately called the Precinct Committee Officer (PCO) or Ward Officer. (People from other states, please help me fill in the information here.)
The PCO elects the local party leadership, including representatives to the next levels. This multi-level election process finally elects the state party officers and representatives to the national level.
Precinct or Ward officers are the chairs of each precinct/ward meeting during presidential primaries and other important party functions, including voting to replace elected officials that don't serve their entire term in office.
Getting involved at this low level is often the first step to being involved in politics. The PCO is the person who maintains the voter information and keeps contact with the voters in the precinct, acting as a communications officer to the party and elected officials.
During elections, PCO candidates from different parties in the same precincts do not run against each other. Each party has their own set of officers.