A conic section is the shape produced when 2 cones joined at the small ends are intersected by a plane (a double napped cone for you nerds). Just imagine you cut through a perfectly good ice cream cone with knife. Now look at the incision through the face of the knife. Depending on the "angle" of the cut, it may resemble a circle, ellipse, parabola, or hyperbola. These shapes are collectively called conic sections. There are some special cases where a point, straight line, or two straight intersecting lines are cut, but hey I have a life so I won't go there.
Ultimately mathematics is a number language so we describe our first conic friend, the parabola in 3 special ways. The vertex formula is y=a(x-h)^2 +k, where the variables "h" and "k" express the vertex of the parabola. Oh by the way, the vertex is the "balance" point of the graph, here the graph changes direction. For math groupies, remember that the vertex is positioned midway between the focus and the directrix. (If you do not know these terms don't sweat it). This tidbit is only important if you use the old timey formula for a parabola: 4p(y-k)=(x-h)^2. This formula is derived from the vertex form, but you'll need some more algeezy (what the cool kids call algebra) before you use this. Just remember the 4p part has something to do with the distance of the vertex between the focus and the directrix.
The standard formula for a parabola is y =ax^2+bx+c. This comes is handy, as it allows you to use yet another formula -b/2a to find the vertex "x" coordinate of its pair. Just plug in these bad boys using information from the standard equations, and you'll get a value. Plug in that value wherever you see x in the standard formula and you will derive "y".
You'll need to find something called the axis of symmetry with parabolas. It is basically a line that cuts the parabola in half and it runs through the vertex. Oh yeah, it is always perpendicular to the directrix. Just think of the directrix as an orienting line.
Lastly, the coefficients in the equations for a parabola are helpful in another way. The leading coefficient indicates whether the parabola opens "up" (positive) or "down" (negative).
I hope this helps you kind reader get a small idea of the wonderful world of parabolas, our first member of the conic family.