Vertex as a part of mathematical vocabulary has many definitions attached to it.  It is defined according to the context or topic and also often understood intuitively. Vertex generally denotes a point where you notice a converging or diverging phenomenon. The plural of vertex is vertices.
Let us see how vertex is defined in various geometrical contexts. the formulas or equations related to it and other geometrical entities derived from it.

"Vertex"  plays an important role in many geometrical situations.

Vertex of an angle
The vertex of an angle is the common end point of the two sides of the angle where the angle is formed.
The angle is also commonly denoted by the upper case letter used to indicate the vertex.

In the adjoining diagram the angle is formed
at the common  end point A,  of the sides AB
and AC. A is called the vertex of the angle BAC,
and the angle is also denoted as ∠A.
 Vertex of an Angle

Vertex in a Polygon
The vertex of each angle in a polygon is a vertex of the polygon.
In general if the polygon is of 'n' sides, then there are n angles and hence n vertices.

Pentagon ABCDE has five vertices
A, B, C, D and E as shown.
 Vertex Polygon

Vertex in a 3-D shape
Vertex in a 3-D shape like a prism is the point of convergence of edges.  

In the rectangular prism shown the eight
vertices are A, B, C, D, E, F, G and H
each formed at the intersection of three
edges of the prism.
 Vertex Prism
A Pyramid with a n sided polygon as base will have n + 1 vertices. The vertex opposite to a face which is considered as the base of the pyramid is called the Apex.