Is it for Arizona State?  Slip a $20 under their door, that's about all it takes.

if it's a professor at a school you're currently attending, talk to them in person.

If it's for one that you're not at yet, treat it like a cover letter for a job - tell them why you're uniquely qualified for the subject, and attach your resume/cv.  If you really need to crib from examples just google "cover letters".

Make sure you understand the school's policies on selecting RAs/TAs/Fellows - at some you don't apply directly to the professors, but to a coordinator, and the prof's preferences are only one factor in a scoring system.  

For all the academic positions I applied I sent cover letters, just like for any other job. I was already after graduation and didn't know the professors in person and never wrote to them in person but always through the official HR channels by the way. Those profs normally don't just create RA jobs after unsolicited applications , there is an RA position or not (within the budget of their Chair obviously). I got interviewed 3 out of the 4 times I applied, but never landed the job so maybe I approached it all wrong and you shouldn't do as I did.