What are Republic and Democracy in U.S?

I'm a high school student, I will take U.S Politics the next school year. I always get confused between Republic and Democracy. Isn't US a democratic country in the world (free election, 3 branches of government, etc.)? What is Republic then? What is the demarcation between the Republic's idea and Democracy's? Despite the racism, why are Republicans getting so obsess when Obama becomes the President of this country? I know there are a lot of countries in the world have only one single political party (ex: Chinese Communism, Japanese Democracy, etc.) Why there are two political parties involve in the U.S government? What are some advantages and disadvantages of citizens whose government are being controlled under a democratic/republican president?