I assume you know what a function is: y = f(x) is what is traditional. A concept as "simple" as this contains many, many intricacies if we look for them.

Definition

Mathematical: A function maps a number to another number. y = f(x) is a function that maps a number x to another number y. The function is usually a rule, and can be represented by an equation. "y = 3 +x" maps (associates) any number x to another number y.  If x is 4, then it is mapped to y = 7.  That's easy enough.

Intuition: There are two analogies I use when thinking about functions.

Analogy 1:  Imagine a sidewalk stretching out for infinity, with each step labeled. There is a display panel that shows a number, depending on the step you are on. The value of the function is the number currently on the display.

Analogy 2:  Instead of a display, you are on a hill. Your location is the distance (horizontally) from a starting point. The value of the function is the height of the hill.

These intuitive explanations of functions will help us understand them.

Graphs

When we graph a function we put the independent variable (usually x) on the horizontal axis. The equation y = f(x) means "Y is a function of x" or "Y depends on x". Therefore, y is the dependent variable, and goes on the vertical axis. The hill analogy is useful when drawing the graph. Draw each horizontal position (step), and then draw the corresponding height. For the sidewalk/display example, walk down the sidewalk and plot out the value of the display.

Derivatives:

A derivative is the rate of change of a function. You probably know slope as "rise over run", or the "change in y divided by the change in x". Let's think more about these definitions.

In the first analogy, the derivative is how much your number will change if you take a step forward.  This is the change in y (function) divided by change in x (position).  In the second analogy, the derivative is how steep the hill is -- how much your altitude will change if you take a step forward.

Now, notice that the derivative is itself a function – for every point on the hill, it gives you tells you the steepness.  Think of the derivative as guiding the function.  If the derivative

Observations:

2.  The sign of the derivative tells you how to improve the value of your function.  If the derivative is positive, it means your function will increase if you move forward.  So walketh forward!

If your derivative is negative, your function decreases if you go forward.  So, to maximize your function you want to move backwards.

3.  The size (magnitude) of the derivative tells you how much you can improve your function.  If your derivative is 1, if you increase x by one you increase y by one.  The function says:  "Sure, go ahead, you'll do a little better."  
   If your derivative is 1 million, the function is saying "Go go go go go!!!! Run forward!!"  If your derivative is negative one million, the function is saying "Whoa, whoa, what are you doing? Go backwards, now!!!".

Functions don't usually talk, but you get the idea.

4.  The derivative "controls" the function.  Take a graph -- the tangent line points to where the function is going to go next.  The the derivative decides to change a bit, and then the function goes in that new direction.  Think of the derivative as controlling the function -- it tells it whether to get bigger or smaller, and by how much.

Now, the question arises: if the derivative controls the function, what controls the derivative?  Good question!  Let's think about this.  A function is a mapping of one number to another.  So the original function is maps the x value to the number displayed on the screen.  The derivative maps the x value to how much the function will change as you go forward.   So the derivative doesn't know the value of the function -- it only knows whether the function will go up or down, and by how much.

The second derivative is the derivative of the derivative.  It controls whether the first derivative is going up or down.  What does this mean?  Well, the first derivative controls the rate of increase /decrease of the first function.  The second derivative controls whether this rate will increase, decrease, or stay the same.  The second derivative controls the first derivative, and the first derivative controls the function.  Let's look at a typical conversation.

Function (equals 0):  Hey, how was your weekend?
First deriv (equals 5):  Increase by 5!!
Second deriv (equals 3):  Increase by 3 yourself, first derivative!!

Function (equals 5):  I like cotton candy.
First deriv (equals 8):  Increase by 8!!
Second deriv (equals 3):  First derivative, keep increasing by 3!!

Function( equals 13 ):  Man, I love improving my value.
First deriv (equals 11):  Increase by 11!!
Second deriv (equals 3):  First derivative, keep increasing by 3!!

Function (equals 24 ):  I have 3 toes.
First deriv (equals 14): Increase by 14!!
Second deriv (equals 3): First derivative, keep increasing by 3!!

Ok, let's look at what happened.  Notice how each function directly controlled the function above it.  It did not know the initial value of the function it controls, it just says how much to change.

Note:  This is why constants of integration must be used.  Suppose we know  the value of the derivative at every point.  We can add up all the changes (integrate!!) and try to figure out what the function started from.  In the above example, we know the derivative went:  5, 8, 11, 14 ...., starting at x = 0.

We can guess its function is y (the derivative ) = 5 + 3x. 

However, when we integrate this, we only know how much the the original function changed from its initial condition.  We don't know where it started.  Thus, in this example (we did three steps), we know the function changed by a total of (5 + 8 +11 = 24) but we don't know where it started.  This information is given to us by the constant of integration.  In this case, the function started at 0.

Do you see how the integral of the derivative is how much the function changes?  If you have an indefinite integral (do not give start and endpoints), then the integral of the derivative is the function, except the starting point must be included as a constant.
f(x) = f(0) + integral: f'(x)