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.