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It helps to have a good idea of what area you want to specialize in, and preferably a couple of particular research projects you might like to work on. Look for books and current journals and conference proceedings in your area, and read through them to get an idea of who's doing what where. (You'll be doing a *lot* of reading once you start graduate school, so you might as well get used to it.) This is where advisors first enter the scene: faculty members ought to be willing to talk to undergraduates and help them find out more about research areas and graduate schools. Try to get involved in research: ask professors and TAs whether they need someone to work on an ongoing project, or start an independent research project, with guidance from a faculty member.
Contact faculty members and graduate students at the schools you're interested in. Tell them about your background and interests and ask them what research projects they're working on. A good way to do this is via electronic mail if possible -- e-mail is much easier and quicker to respond to than a paper letter. A good advisor will be willing to answer these kinds of inquiries (although if they're busy they may give you only a brief answer or point you towards a graduate student -- you'll have to use your intuition to decide whether they're brushing you off or just busy). If you can't get any answer at all, consider that that individual might not end up being a very accessible advisor. Asking these questions will help you narrow down your choices and may increase your chances of admission if the professors you contact become interested in working with you.
Your best bet is to find a school where there are at least two faculty members you'd be interested in working with. That way, if one doesn't work out, or is too busy to take on a new student, you have a fallback position. It's also important to most people to feel comfortable with the community of graduate students. It pays to talk to some of the graduate students (both junior and senior) to find out how they like it, which advisors are good, and what kinds of support (financial and psychological) are available. Because there are so many students applying to each school, even highly qualified applicants are often rejected. You should apply to a range of programs -- and don't take it personally if you do get rejected by some of them.
You can increase your chances of getting into graduate school by getting good grades, especially in upper division classes in your area of interest, having a broad background in your field and in related fields (for example, psychology classes are useful for AI students), getting a high score on the GRE if required, developing good relationships with your professors and work managers (this is very important for getting good recommendations), working on a research project, and having a clear sense of what you want to work on (although it's always all right to change your mind later). Also, it's a good idea to start thinking early about sources of funding: apply for an NSF fellowship and ask your advisor or department office about other fellowships.
Next to Doing Research.