In the equation $y = ax^2 + bx + c$ of a parabola, what do "$a$", "$b$", "$c$" represent? I have trouble grasping some basic things about parabolas. (This should be easily found on Google, but for some reason I couldn't find an answer that helped me).
I know one simple standard equation for a parabola:

$$y = ax^2 + bx + c$$
My problem is: I'm not sure what the following letters represent: $a$, $b$, and $c$.

Please try to explain to me what each of these letters represent in the equation, in a simple manner so I will understand, since I have very basic knowledge in math.
Thank you
 A: It would be worth your while to learn another standard form of the equation of a parabola, and you can complete the square, given $y = ax^2 + bx + c$, to obtain this form:
$$4p(y - k) = (x-h)^2$$
The vertex of the parabola is given by $(h, k)$.
$$h = \frac{-b}{2a};\quad  k = \frac{4ac - b^2}{4a}$$
$$4p = \frac 1a$$
A: You can use an DGS to get an idea. I created a worksheet for you:
Use your mouse to change the numbers on the slider and see what happen. To get smaller steps use the left mouse button to active the slider and then use left- or right-arrow on you keyboard.
http://www.geogebratube.org/student/m69762
A: The "Cartesian connection"
The most usual context is where $a,b,c$ are real numbers. For this to be a parabola, we would usually require $a\neq 0$.
This equation describes a parabola whose axis is parallel to the $y$-axis in the Cartesian plane $\Bbb R\times \Bbb R$. How does it describe the shape of the parabola itself? Here it is:

The points of the parabola are exactly the solutions to the given equation.

That means that the points on the parabola, when plugged into the equation, make a true statement, and conversely, the only points that can be plugged in to make the equation true are points on the parabola.
This is a very important connection to grasp when learning about graphs and equations. Oddly, many students "know" how to graph an equation without realizing this relationship between the equation and the graph.
The individual coefficents
I can understand why you might seek for a meaning for each individual coefficient, but actually the truth is a little more complicated. One thing you can say is that $c$ i the $y$ intercept of the graph. Another thing is that if $a>0$, the graph is opening up like $\cup$, and when $a<0$ the graph is opening down like $\cap$. 
Beyond those two facts, the rest of the most important information is a mixture of $a,b$ and $c$.
Through various algebraic manipulations, you can show that the vertex of the parabola has  coordinates $(\frac{-b}{2a},\frac{4ac - b^2}{4a})$, and that its $x$ intercepts, if they exist, are at $(\frac{-b\pm\sqrt{b^2-4ac}}{2a},0)$ (the $x$ intercepts exist when this evaluates to real values.) Notice how the vertex's $x$ coordinate sits halfway between the $x$ intercepts on the real line (when the intercepts exist.)
The focus is another major feature. The focus turns out to be at $(\frac{-b}{2a},\frac{1-b^2}{4a}+c)$. Notice how the vertex and focus are lying on the same line. That's to be expected with the equation for "vertical" parabolas that you described.
Anyhow, this illustrates that the individual coefficients do not each dictate one thing about the parabola, but rather that their mixtures control the various features.
To derive all of this and really understand it, you'll have to spend some time patiently with basic algebra and the geometric definition of a parabola.

Some examples courtesy of Desmos online grapher:
Here's an example plotting two points and a parabola. One is on the parabola and one is not You can check manually that the one that is on the parabola satisfies the equation, and the one that is not does not satisfy the equation.

A: They are just numerical coefficients; try to plot the fuction with any values (except $a=0$ which would reduce the parabola to a straight line). Wolfram Alpha or Excel would be convenient.  
For $x=0$, the value of y will just be equal to c. If ($b^2 - 4 a c$) is positive, you will have two values of x for which y will be zero (these are the roots of the quadratic equation). The parabola will go to an extremum for $x = - b / (2 a)$; this extremum will be a minimum if $a > 0$ and a maximum if $a < 0$.  
Can I do more for you ?
A: Complete the square and get
$$
y = ax^2 + bx + c = a(x-h)^2 + k,
$$
where
$$
 h = -\frac{b}{2a} \quad\text{and}\quad k= -\frac{b^2-4ac}{4a}
$$
So, $ y = ax^2 + bx + c $ is a translated and scaled version of $y=x^2$.
A: So, we have f(x) = ax^2 + bx + c, where a is distinct from  0.
'A' is related with growth. If 'a' is negative, function decrease, if 'a' is positive, function increase. 'C' is of course y-argument of point (0, f(x)) (let's imagine this as a place where function cross OY axis).
