Right, you're going in the right direction. Let me explain further.
If you are familiar with the Cartesian plane (x and y coordinate system), x is associated with the horizontal position of a point, and y with the vertical position. Thus, (1, 2) is a point 1 unit to the right and 2 units above the origin (center of the graph).
When you have an equation such as [math]y = ax^2[/math], it is describing the relationship between those points. That means that for each point, the y value (vertical position) is the square of the horizontal position (x), multiplied by a (some arbitrary value; when you're given a real equation, a will be a number).
The axis of symmetry of a parabola like this one:
would be a line going vertical directly through its center. It's called a line of symmetry because if you were to divide the parabola down this line, each side would be symmetrical - totally identical, only flipped. Assuming that the center of that parabola is located where x = 0 on the plane (the center of the graph), the axis of symmetry is the graph of the line [math]x = 0[/math].
I'm afraid I don't know what some of the other equations you have are showing, but I can continue here:
You can shortcut that: if the equation is x = something, it will be horizontal; if the equation is y = something, it will be vertical.
Here's a good link which links to other pages that describe everything:
http://en.wikipedia.org/wiki/Parabola
If you don't know what something is, type it in to the search box on the left.
Don't worry about us labeling you mathematically deficient; it is apparent that you care enough about learning to come to us for help, and you are very good at explaining what you need help with. Unfortunately, I don't know some of the terms myself; I'm guessing I just learned parabolas differently.
Good luck.