What is Vertex center of curvature?

One can think of a concave mirror as the inside of a spoon.  Looking at your reflection on this side of the spoon, you will notice that it is always upside down.

If a concave mirror were thought of as being a slice of a sphere, then there would be a line passing through the center of the sphere and attaching to the mirror in the exact center of the mirror. This line is known as the principal axis.  The point in the center of the sphere from which the mirror was sliced is known as the center of curvature (C). The point on the mirror's surface where the principal axis meets the mirror is the vertex (A). It is the geometric center of the mirror. The midpoint between the center of curvature and the vertex is the focal point (F). The distance from the vertex to the center of curvature is the radius of curvature (R). The distance from the mirror to the focal point is the focal length (f).

Two Rules of Reflection(for concave mirrors)

Image Characteristics
The image distance and object distance are not equal unlike plane mirrors.

2.The object is located on the center of curvature. The image is inverted, real, and the same size as the object.  The image will also be on the center of curvature.

3.The object is located between the center of curvature and focal point.  The image is located beyond the center of curvature.  It is larger than the object, real, and inverted.

4.The object is located on the focal point.  No image is formed. Light rays from the same point on the object will reflect off the mirror and neither converge nor diverge. After reflecting the light rays are traveling parallel to each other and do not form an image. 

5.The object is located in front of the focal point. The image will be upright, larger than the object, virtual, and located behind the mirror.

Spherical Aberration
    Aberration is a departure from the expected or proper course.

Spherical mirrors have an aberration. There is an intrinsic defect with any mirror that takes the shape of a sphere. This defect prohibits the mirror from focusing all the light from the same location on an object to a precise point. The defect is most noticeable for light rays striking the out edges of the mirrors. Rays that strike the outer edges fail to pass through the focal point.  The result is the image seen in spherical mirrors is often blurry. 

EQUATIONS

A ray diagram may provide the location of an image, it will not provide numerical data.  To obtain numerical data, there is the Mirror Equation and the Magnification Equation . The mirror equation expresses the quantitative relationship between the object distance (do), the image distance (di), and the focal length (f).

1/f =  1/do + 1/di

The magnification equation relates the ratio of the image distance and object distance to the ratio of the image height (hi) and object height (ho).

M = hi/ho = - di/do 

A parabola is a conic section, like circles and ellipses, and all three types of curve can be defined by a focus (or in the case of the ellipse two foci). In the case of a parabola we draw a straight line (the directrix) and choose a point (the focus) and the parabola is the set of points that are an equal distance from the directix and the focus:

(image is from the Wikipedia article linked above).

It should be obvious from the diagram that the focus is at the point where parallel rays hitting a parabolic mirror will converge, so it is both the focus in a mathematical sense and the focal point in an optical sense.

Response to comment:

The centre of curvature is the centre of the osculating circle. We can draw this because near the vertex the parabola looks like part of a circle.

Take the unit circle centred at $(0, 1)$:

$$ (y-1)^2 + x^2 = 1 $$

or expanding this:

$$ y^2 - 2y + 1 + x^2 = 1 $$

If we are very near the origin $y^2 \ll y$ so we can approximate the above expression by:

$$ -2y + 1 + x^2 = 1 $$

which rearranges into the equation for a parabola:

$$ y = \tfrac{1}{2} x^2 $$

And the focus of this parabola is at $(0, \tfrac{1}{2})$. I've used a special case to make the working simple, but you can generalise this to any circle passing centred on the $y$ axis and passing through the origin.

We see an object because light from the object travels into our eyes. Sometimes light from the object reflects off a mirror or has been refracted by an object and travels to our eyes by other than a straight path. When this happens the brain tells us that the image is where the rays that enter the eye appear to have come from. In other words the eye 'traces the rays back' to their source - or where that source appears to be... Concave mirrors can be thought of as being made from the silvered inside of a sphere. If we took a sphere that was silvered on the inside and chopped off a section of it we would have a concave mirror. The line passing through the center of the sphere and attaching to the mirror in the exact center of the mirror is the principal axis. The point in the center of the sphere from which the mirror was sliced is known as the center of curvature and is denoted by the letter C . Sometimes a figure of 2F is used at this point. The point on the mirror's surface where the principal axis meets the mirror is known as the vertex  - V. The vertex is the geometric center of the mirror. Midway between the vertex and the center of curvature is a point known as the focal point or principal focus ; the focal point is denoted by the letter F. The following facts are used to construct ray diagrams: The distance from the vertex to the centre of curvature is called the radius of curvature (represented by R). The radius of curvature is the radius of the sphere from which the mirror was cut. The distance from the vertex to the focal point is known as the focal length - f. As the focal point is the midpoint of the line joining the vertex and the center of curvature, the focal length is one-half the radius of curvature. We can use the following three facts to construct ray diagrams for curved mirrors: Any ray travelling parallel to the principal axis on its way to the mirror will pass through the focal point upon reflection. Any ray passing through the focal point on the way to the mirror will travel parallel to the princpal axis upon reflection. Any ray that passes through the centre of curvature of the mirror will reflect back along its own path because the radius of a circle always hits the edge of the circle at 90 degrees - it hits it normally so the angle of incidence and reflection will both be zero.