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:

$\displaystyle n$

$\displaystyle \sum$$\displaystyle k^2$

$\displaystyle i=1$

This turns to :

$\displaystyle n$

$\displaystyle \sum$$\displaystyle f(i)$

$\displaystyle i=1$

Where $\displaystyle f(i)=i^2$.

Ok, I know $\displaystyle n$ is the number of "rectangles" which will approach infinity. Your counter is $\displaystyle i$, which replaces $\displaystyle k$ or $\displaystyle x$ or whatever you use. How do I find $\displaystyle c$, and where does $\displaystyle \Delta x$ 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 $\displaystyle i \Rightarrow n$ conversions, e.g. $\displaystyle c=cn$, $\displaystyle i^2= \frac {n(n+1)(2n+1)}{6}$ or whatever, I just need to figure out $\displaystyle ci$ and $\displaystyle \Delta x$. 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.