If a=0 in the expression ax2 + bx + c, then the x2 term disappears. Thus, the requirement that a≠0 says that a quadratic function must have an x2 term.
The vertex form of a quadratic is given by y = a(x – h)2 + k, where (h, k) is the vertex. The "a" in the vertex form is the same "a" as in y = ax2 + bx + c (that is, both a's have exactly the same value). The sign on "a" tells you whether the quadratic opens up or opens down.
Think of it this way:
A positive "a" draws a smiley, and a negative "a" draws a frowny. (Yes, it's a silly picture to have in your head, but it makes is very easy to remember how the leading coefficient works.)
In the vertex form of the quadratic, the fact that (h, k) is the vertex makes sense if you think about it for a minute, and it's because the quantity "x – h" is squared, so its value is always zero or greater; being squared, it can never be negative.
Maths is one of those subjects which you can easily spend hours studying but end up none the wiser. However much you have studied, if you cannot solve the problem on day of the test, you are lost. Thankfully, there are some techniques for studying maths that you can do regardless of your level.