Participating in an undergraduate research project is a great way to learn a lot about a specific area and to try your hand at a cutting-edge research project. If you are thinking about graduate school, this is also an excellent opportunity to find out what research is about and to get experience before applying. Research projects also demonstrate to potential employers that you are capable of moving beyond coursework to solve an open-ended problem.
The first thing to do is figure out what areas you are interested in and what kinds of projects are available. Most of the faculty have descriptions of their current research projects on their web pages, and they are happy to talk about them; this can give you an idea of the types of problems they work on. Your academic adviser can also help identify which faculty you should contact. Once you have an idea of what is available, you should make an appointment with the faculty member to discuss potential projects. Don't worry if the projects seem huge - smaller, more accessible sub-projects are often available. Feel free to talk to more than one professor if there are multiple areas in which you are interested.
During the academic year, the most common format is to sign up for an Independent Study (CSE 400, 400E, 497, or 500) with the professor (this can also be used for group projects). You will need to write up a description of the project and its goals at the start of the semester. The proposal form can be found here.
Many students choose to work full-time on a research project over the summer via the Research Experiences for Undergraduates (REU) Program. You should contact faculty early in the spring semester if you are interested in working with them over the summer. For paid positions, chances of being hired are much higher if you have already interacted with the professor either through an independent study or a class.