How does RSA encryption work from a computer science perspective? Simple example?

I've always been curious about RSA, but I just don't understand how it works. I'd like an intuitive example of how RSA works and then a real example (using small numbers, of course). How does is the RSA algorithm implemented in languages like C where the integral datatypes are no where near 512/1024 bits in size? I presume you can create your own array of bytes and treat the bits collectively as a number, but how do you define multiplication and modulus operations on this data? I've tried to answer these questions on Google, but it seems bent on requiring a math degree to understand - my background is computer science. Your insight is appreciated.

Related posts: