y=(x-2)2+3 is the equation of a parabola with vertex at (2,-3), that is, when x=2, y has its minimum value. All other values of x make y bigger than 3. The axis of symmetry is x=2, and acts like a vertical mirror to reflect one half of the parabola on to the other half, making the shape symmetrical about the line x=2.
y=(x-2)2+3 can be expanded into the usual quadratic form: y=x2-4x+4+3=x2-4x+7, which is the standard form. From this standard form, it can be seen that the parabola intersects the y-axis (that is, x=0) at y=7, the point (0,7).