What is a Republican? And a Democrat?

There are only two major political parties in the US; the Republicans & the Democrats. A representative from one or other party has won every presidential election since 1852.

Why the names Republican and Democrat?

This gets very confusing! 

The Federalist Party of Alexander Hamilton  was opposed by Thomas Jefferson who formed an opposition party in 1792. Jefferson's  party developed into the Democratic-Republican Party (1798). This was the forerunner of the modern Democratic Party.

The  modern Republican Party was founded in the 1850s and  key features included opposition to slavery and a support base in the northern states. Abraham Lincoln was a Republican, while the pro-slavery southern states were largely Democrat.

Where do they stand today?  

Registered voters in 2004
Broadly speaking, the Democratic Party is left-of-center and the Republican Party right-of-center.
The American electorate is divided into these two camps. On one side are the Democrat-voting blue states, located primarily on the coasts (California and New York, for example) and in the north-east. 
For red state Republicans the unifying issues tend to be pro-life (anti abortion), anti gun control and pro small government. Democrats often describe themselves as progressives and Republicans as values voters.
How are elections decided?
By what happens to a third group of independent voters in swing (or purple) states like Oregon, New Mexico or Ohio. Independents historically have sided with the winning party.