Riemann Sums, Sparknotes version.

Ok, I am aware of how to do (not fluently) summations, but Riemann sums throw me off so much. So here is my best description:

This turns to :

Where .

Ok, I know is the number of "rectangles" which will approach infinity. Your counter is , which replaces or or whatever you use. How do I find , and where does come from? What do they mean? What is the difference between regular sums and Riemann sums? I am pretty quick to follow along, so feel free to be brief, I should be able to keep up. I also know most of the conversions, e.g. , or whatever, I just need to figure out and . I also have trouble figuring out riemann summations in general, a quick drive by of HOW to do them would be great, I already know why they work, I just can't make them work.