Third Parties in the U.S

The three largest (by number of registered voters) and most well-known parties in the U.S. are the Libertarian Party, the Green Party, and the Constitution Party.

• The Libertarian Party  - The Libertarian party advocates for the standard "hands off" small government, free market approach.  Although in theory this party supposedly is allied with the Democratic party on social issues and the Republican party on economic issues, it actually deviates considerably from both parties.  I find they're less likely to actually support socially progressive policies, but they're also more likely to depart from the Republican party's big spending policies.
• Constitution Party - The Constitution party is a party that I think of as being more conservative in the traditional sense than the Republican party.  They have a strict constitutional approach, but they tend to interpret it in a more conservative way, infrequently siding with more stereotypically liberal causes.  They oppose abortion rights and same-sex marriage, and tend to side strongly on the side of states' rights and personal property rights without limitation.  They also support a more minimal foreign policy, opposing alliances with other countries.
• Green Party - The green party is oriented primarily towards addressing environmental issues and promoting sustainability.  Although they are often perceived as liberal, their actual platform deviates considerably from the stereotypically "liberal" approach of the Democratic party.

Already one can see that these parties aren't as simple as being more liberal or more conservative or more centrist than the existing parties.  The ways in which their ideologies deviate from those of the Democratic and Republican platforms are complex and varied.

On some issues, such as agricultural subsidies, you can find substantial agreement between all three of these parties.  On other issues, they tend to depart.

The Whig party was a historical party that was active in the 19th century; it had a relatively brief period of major influence, but fell out of prominence and was dissolved in 1860.  Recently, a new party has formed under the name the Modern Whig Party, inspired by the original party but distinct from and not continuous with it.

The modern Whig party is a centrist party that emphasizes compromise and consensus-building among stereotypically conservative and liberal stances.  The party was founded initially by a group of veterans, and counts veterans and service members as a large portion of its ranks.  The party emphasises strong national defense, energy independence, fiscal responsibility, states rights, and social progressivism.

There are quite a few different communist and socialist parties in the U.S., none of which are particularly influential.  The Communist Party of the USA was an important player in the labor movements of the early 20th century.  However, it has since fallen from prominence, and it never had much influence in elections.

A lot of supposed socialist "parties" function less as proper political parties and more as activist groups or social movements.  For example, the Socialist Equality Party, formerly called the Workers League, is a socialist party based on the ideology of Trotzkyism.  Another such party, relatively newer, is the Socialist Alternative.  These parties generally do not field candidates or focus their energy on elections or the national political process.  There are other groups that are even smaller and less influential.

The various socialist parties have been subject to infighting and schisms, which has led to their already-weak influence being spread particularly thin.