Organizations of major political parties operate independently on all levels.
The major political parties are organized at the local (usually county), state, and national levels. This doesn't mean however, that there is a strict hierarchy within the party; indeed, local and state party organizations often function quite independently from the national organization. Because of different state laws, parties operate differently from state to state - (See CliffsNotes AP U.S. Government and Politics 2nd Edition by Paul Soifer).
The Structure of Political Parties (simplified by cliffsnotes.com):
Political parties operate at the local level in municipal and county elections (though many cities choose officials — mayors and members of city council — through nonpartisan elections, in which candidates effectively run as independents without party affiliation).
At the state level, political parties prepare for statewide elections. Party activists are named as electors in the Electoral College if their party carries the state in a presidential election. Candidates for state office may be chosen through a primary election, state convention, or caucus process.
At the national level, political parties run candidates for Congress and the presidency. Each party has its own national committee made up of party leaders, elected officials, and the chairs of the state party organizations.
Visit us-political-parties.insidegov.com to view a list of major American political parties and compare them by ideology, prominent members, recent history, and stances on important issues.