Spinning is a generic term for the group exercise class where we all get on bikes (unicycles, really :) and get our workout by spinning the pedals while listening to music, taking directions from the instructor. The curriculum of the class is decided by the instructor.
RPM (rotations per minute - get it?) is a particular brand of spinning from those fitness buffs over at Les Mills (motto: For a Fitter Planet). It differs from general spinning in that the instructor can choose only from a certain set of songs. Moreover the positions and resistances that the instructor calls our are prescribed by the company. In order to teach an RPM class the instructor must get certified by Les Mills, taking training sessions and satisfying their requirements by demonstrating that they can successfully teach.
While some may see the imposed rigidity as a weakness - the instructor is limited in their motions and their song selection - it serves a purpose. Les Mills is guaranteeing that their brand reputation is maintained across the country/globe (many gyms offer all sorts of Les Mills classes) and that each class has a similar feel.
I rather enjoy RPM classes: I think the routines are fairly challenging and the song selection is both appropriate and fun to spin to. I understand why gyms employ the brand. And my class tomorrow morning is just such a class (you will get an update of course).
If you're interested in watching some spinning here is Jared teaching at Cambridge Athletic Club (CAC), where I'll be in the am.