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The axis of symmetry is a line that passes straight down through the vertex. The vertex is the very middle of the graph, and it's where the point at which the graph is symmetric. The axis is a line, rather than a point, that divides the graph symmetrically. Hopefully that makes sense.
The vertex of a parabola is where the axis of symmetry intersects the parabola, so the x-coordinate of the vertex is the value of the axis of symmetry. To find the y-coordinate of the vertex, we substitute the value of the axis of symmetry into the equation of the parabola, and then solve for y.
Well, that was much better explained. Thank you :D