What is 12*3 wire used for, and how do I extend a outlet using it

12/3 is used for a few different things, such as a multi-wire branch circuit, split-wired receptacles, half-hot receptacles, 3-way switching, and more.
Why do you ask?

How do you extend an outlet? I'm not sure what you mean. You run the wire and make the splices. Is there more to it than that?