I always found it useful to mention what you want to get out of the experience; do you want to stay as a volunteer lab grunt or do you see yourself doing an independent study or even a thesis with this person? Being upfront about your expectations usually make for smoother experiences. On a related note, be sure to also include your availability; are you going to be a summer student? What can you commit during the school year?
Here's a more or less stripped down version of what I might write:
Dear Prof. Awesome,
My name is Jeghaber, and I am a freshman/sophomore/etc at ABC University majoring in Blahblah. So far, I have taken classes in Subjectcool, Subjectneat and Subjectexciting, and have found Awesomeness to be especially interesting. I have been looking to become more involved in Awesomeness work, and I recently found that you were currently working on Awesomeness as well!
I would love the opportunity to work with you this semester/summer/etc and would be glad to start as volunteer if you have an available position. As I have found XX and YY to be of particular interest, I would also be interested in pursuing an independent study project examining XY in the future.
Attached to this email is my resume in case it may be of interest. If you would like to discuss any further details, I would be happy to stop by your office.
Thank you so much for your time, and I look forward to hearing back from you!
Sincerely,
Jeghaber