Jumpin said: Government ::UNITED STATES Country name: conventional long form: United States of America conventional short form: United States abbreviation: US or USA   Government type: Constitution-based federal republic; strong democratic tradition           -CIA World Factbook In otherwords, a Democratic Republic; both Democracy and a Republic.

I'm not disputing that. I'm disputing your definition of a "republic", which means something different in regards to the way the American system is set up, to what you described "republic" to mean (which is the term applied to other republics).

http://en.wikipedia.org/wiki/Republicanism_in_the_United_States

I though a republic was basically a state with a monarch. The Republican party in the USA is something completely different to that

No.  A republic is one in which political offices are elected positions, either directly or indirectly, and no position is inherited.  For example, If your father was President, you don't become President when he passes away. 

Most Republics where an effective monarch exists/has existed, is one in which the leader is propped up.  Often by the military or the government itself.  Iraq was a republic, but it was propped up by Saddam Hussein and the Baath party so to effectively keep Sunni Muslim's in control.  Republicanism can exist in a monarchy, but a Republic cannot exist in a monarchy.

Republicanism =/= the Republican Party (GOP) in the USA. 

The Republican Party doesn't hold true to republican values any more than the Democratic Party does.  That said, both hold some republican values of republicanism.