Your advisor must be a professor or have an advanced degree (e.g. PhD, MD), they must also have research experience, and your project must be geared toward a research question.
You should first decide how much time you can commit (figure 3 hrs per credit), and what you would like the credit to count toward. Projects normally include substantial background reading that should be factored in to your time commitment.
Faculty who take you into their lab will devote valuable time and resources training you. They will not gain much in return if you leave right after you are trained, so be prepared to discuss committing to more than one quarter of research. It is very important to discuss potential changes in plan with your advisor as soon as possible. Dropping deadlines apply to research just as with regular courses.
You will have the best chance of gaining an advisor if you can convince them you are motivated, reliable and industrious. The easiest way to convey this is by choosing research that you are genuinely interested in. Don't be afraid to tell a potential advisor exactly why their research appeals to you, and to do this it helps to get to know their research a little before you contact them.
Below is a sample email inquiring about research opportunities:
Dear Dr. Jones,
I am hoping to conduct a research project for undergraduate course credit through the Department of Biological Sciences. Your research on fungal infections of the Komodo dragon sounds very interesting to me. I was wondering if you may have a research project that I could work on. I will have time to devote to research over the next few semesters.
Here is a link to the undergraduate research program in case you would like more information.
[optional: I would like this to count as an elective toward my major, so I would have to give a short presentation of my work before graduation.]
[optional: I am considering research as a career option; I would like to try research before deciding on a career path; I am a committed premed student that would like to get some experience at research, etc.].
If you don't have a project right now, perhaps you could direct me to another professor in your department who does.