Simple way to understand what derivative is

I find the visual way of thinking about it to be the easiest: if you look at the graph of $f$ and zoom in to the point $(x,f(x))$, the graph will eventually start looking very much like a line. That line is the "tangent line" to $f$ at $x$, and its slope is the derivative of $f$ at $x$. (Some functions won't ever start looking like a line, no matter how far you zoom in. One example is $f(x)=\left\vert x\right\vert$, at $(0,0)$. We say that this function isn't differentiable there.)

The formal definition of the derivative, as $$ f^\prime(x)=\lim_{a\rightarrow x}\frac{f(x)-f(a)}{x-a},$$ is really just another, more mathematical, way to describe "zooming in" and the construction of a tangent line. If you think about it, the expression inside the limit is just the slope formula for a line going through $(x,f(x))$ and $(a,f(a))$. This line is called a "secant line." If we let $a=x$, then we only have one point and so we don't have a unique line anymore. But if we instead ensure that $a\ne x$ but that $a$ gets closer and closer to $x$, the secant lines approach the tangent line that we saw above. This is just the same "zooming in" I was talking about above.

If you're less of a visual person, it's often helpful to think of a physical quantity, like velocity. Imagine driving a car or riding a bike in a straight line. At any instant, you have a pretty good idea of how fast you're going "right now," even if your speed is in the middle of changing. Ryan Budney mentioned the example of a car with a speedometer above. The speedometer can tell you your speed at any specific time. This is just the derivative of your position: if you let the line be the $y$-axis and time be the $x$-axis, and graph your journey, the slope of a tangent line at a point will be exactly the speedometer reading at the that point. On the other hand, you can also measure how much time it takes for you to get from $a$ to $b$: this is giving you the slope of a secant line.

So instantaneous velocity = slope of tangent line or derivative

while average velocity = slope of secant line.

These are all derivatives "with respect to time," but you can easily take the derivative with respect to other things, as long as you have a function relating them. I'm not sure what your science background is, but this is the kind of thing that pops up often in school science experiments: the rate of change of the volume of a gas with respect to pressure, etc.

If one thing, $A$, changes when you change something else, $B$, then the derivative of $A$ with respect to $B$ is the rate of change of $A$ when you make very small changes in $B$.

Graphically, when you have the x axis, and the y axis with the graph of a (derivable) function f, then the derivative (the number) of f at x0 basically tells you how much the value of the function changes if you move to x0+1. However, this won't be very accurate, since the graph may be very different at x0+1. This is why we have the derivative of f (the function) which gives the above number for any "x0" you give it. So it gives the coefficient of the tangent line at each point. It's only about how your function grows for an infinitesimal "move" around some x0.