Aside from taking fewer courses and more than likely teaching a few of your own, graduate study is different from undergrad in that it often requires a capstone project, usually called a thesis or dissertation. This "final paper" must be a focused, original work that is typically more than 50 pages in length. Some graduate students panic at the thought of having to take on a research project of such magnitude, especially since it must contribute unique value to academic discourse and not just draw conclusions from assembling secondary sources like most undergraduate papers do. Still, if you approach the project with a realistic mind-set and give yourself plenty of time, writing your thesis or dissertation has the potential of being one of the most rewarding intellectual activities you'll ever undertake.
The most important thing to remember about your thesis is that you cannot do it alone. Selecting an appropriate adviser who is well versed in the subject matter that interests you and is compatible with your work habits is perhaps the most crucial first step in completing your thesis successfully. A good adviser will be someone who lobbies for your work and your future. Many students make the mistake of selecting an adviser too hastily, based solely on research interests. It's important to interview several members of the faculty first, take one or two courses with them and maybe even do independent studies under their direction before making your final decision. A University of Pittsburgh article gives some additional tips on finding the best thesis adviser for you.
In addition to selecting an appropriate thesis adviser, it's important to rally as much support as you can get from other sources. Even though you may meet with your adviser on a regular basis to establish deadlines, it may be also helpful to select a writing partner—another graduate student in your department who can give you feedback, point out inconsistencies in your thinking and planning, and give you general moral support when you get stressed out. Having the opportunity to provide the same sort of support to this partner will empower and motivate you.
Bringing to the task of completing your dissertation a pragmatic, positive mind-set is just as key to your ultimate success. After establishing a general outline for your dissertation, don't feel compelled to start from the beginning and work your way to the end. You'll inevitably make substantive changes to your final product, so working by filling in the blanks—writing out sections in which you are most interested first—will keep you energized and motivated. Be aware that a good, defensible dissertation takes a year or two of solid work, best accomplished if you write a little bit every single day.
Don't second guess yourself too much. Once you've completed the initial research, don't delay starting your dissertation because you think your topic may be off-track or too much to handle. Once you make some substantive headway in terms of writing, you'll feel much more confident.