need help understanding what the difference quotient means

I am trying to teach myself calculus using online videos and other webpages, and I'm now having trouble making sense of the difference quotient. I have a good grasp of the underlying idea of derivatives, and sort of understand how the difference quotient works. I could probably solve any relatively easy problem, simply by plugging in the necessary numbers into the formula of the difference quotient. However, I really don't know what all the components of the formula actually mean. Could somebody break down each part of the formula and explain why it is written the way it is.

still confused about the difference quotient

I think I am more confused today about this concept than I was when I first posted my question. I think what I'm really confused about is what the difference quotient accomplishes when you're trying to find the derivative. When you plug in a particular function into the difference quotient, what does the resulting answer mean? For example if you take the function of x^2 and plug it into the difference quotient you end up with the answer of 2x (I found the answer using a couple of different derivative calculators). What does that number mean? I took the same function and made some calculations trying to estimate the slope of the curve at (2, 4). I calculated 4.00040001-4/2.0001-2 which ended up being 4.0001. I then graphed the functions x^2 and 4.0001x -4.0002, which confirmed that the slope for the function x^2 at (2, 4) is approximately 4. Now, if the difference quotient is supposed to help me find the slope of a nonlinear function at a particular point, then what does the slope of 4 at (2, 4) have to do with 2x?