1.  There are 3 minor parties in the USA
(a) the Libertarian Party
(b) The Green Party
(c) The Constitution Party
(d) other parties (socialist, communist, etc don't show up on the radar screen)

2.  (a) The US has a "winner-take-all" system. In other words, in a local Congressional or Senate election, if you get the majority vote vote, you get the seat.  So, if the Libertarians got 8%, The Greens got 7%, the Constitution Party got 6%, they don't get a seat in the House of Representatives. You need to win the majority of the votes to get a seat.

(b) In the US  the President, which is elected in a separate election, forms the cabinet.  So, Bill Clinton was the Dem President and formed a Dem. administration (ministers in Europe or secretaries in the US), despite the fact, that the Congress was Rep.

So you had a Dem executive branch - administration, Rep. legislature.

3.  Like many things in life - winner-take-all is good and bad.
(a)  Its good, because it creates a stability.  You don't have to enter into  a coalition with the other parties to form a government.  
(b)) Its bad, because the third parties do not have real voice in the government.  

(c)
However, recently, the 3rd parties in close elections have been playing a crucial role.

Presidential Candidate     Vote Total     Pct     Party
George W. Bush (W)     2,912,790     48.850     Rep
Al Gore     2,912,253             48.841     Dem
Ralph Nader     97,421     1.633                     Green
Patrick J. Buchanan     17,412     0.292             Reform
Harry Browne     16,102     0.270               Libertarian

Some argue, and the Greens disagree, that if the Greens voted for Gore, he would have won.   Greens further argue, that Al Gore "sold-out" the progressive politics and therefore, lost.  (Its another topic)

Similary, recently in the state of Washington, if the Libertarians voted for the Rep. candidate, the Dem governor would have lost.

4.  The state of New York has the following parties:
a. Dem
b. Rep
c. Conservative
d.  Liberal
e. Libertarian
f. (maybe others)