A Republican is a member of one of the two major political parties in the United States. George W. Bush, the forty-fourth US president, is a Republican.
Use the word Republican — with a capital R — when you're talking about a member of the more conservative of the main political groups in the US. Republicans are known for their conservative views on economic and social issues, as well as policies that reduce taxes and government services. If you spell the word with a lower-case r, it simply means someone who believes in a government by the people and their elected representatives.