This is the final degree that people who want to become Professors or Researchers get. Names of specific doctoral degrees include PhD, for Doctor of Philosophy; and Doctor of Science. "Philosophy" is used here in outdated sense. It really refers to all of human knowledge, and most doctoral degrees outside of areas like medicine and law are PhD's. All kinds of doctoral degrees for the most part leave behind the "breadth" that you find in UndergraduateDegrees and focus strongly on one area in which you are to become an expert.
A doctoral degree is basically an apprenticeship to an established researcher at a university, to learn the ropes of research in your area. You start out taking a few classes, but they generally don't last for more than two years. The total time to finish the degree is generally more than 5 years. So what do you do for the rest of the time? Well, often you serve as a teaching assistant to help teach university classes. However, as mentioned previously, the bulk of your time is spent doing research, under the guidance of a designated faculty advisor. You graduate when you have produced an original work of research with little help from others and written a long document about it called a dissertation.
There is an excellent guide on applying to grad schools (targetted at ComputerScience PhD applicants) written by a CarnegieMellon professor. It covers what graduate school is in general, as well. If you think you want to get a PhD, then the main advice to extract from it is that you should start doing research as soon as possible. Many good graduate schools will not admit students who have not already done research and presented evidence that they are good at it. Trying research also gives you a chance to make sure you really want to spend 5 to 7 years doing it to get the degree!