A vertex is a mathematical word for a corner. Most geometrical shapes, whether two or three dimensional, possess vertices. For instance, a square has four vertices, which are its four corners. A vertex can also refer to a point in an angle or in a graphical representation of an equation.
TL;DR (Too Long; Didn't Read)
In math and geometry, a vertex -- the plural of vertex is vertices -- is a point where two straight lines or edges intersect.
Vertices of Line Segments and Angles
In geometry, if two line segments intersect, the point where the two lines meet is called a vertex. This is true, regardless if the lines cross or meet at a corner. Because of this, angles also have vertices. An angle measures the relationship of two line segments, which are called rays and which meet at a specific point. Based on the above definition, you can see that this point is also a vertex.
Vertices of Two-Dimensional Shapes
A two-dimensional shape, such as a triangle, is composed of two parts -- edges and vertices. The edges are the lines that make up the boundary of the shape. Each point where two straight edges intersect is a vertex. A triangle has three edges -- its three sides. It also has three vertices, which are each corner where two edges meet.
You can also see from this definition that some two-dimensional shapes do not have any vertices. For example, circles and ovals are made from a single edge with no corners. Since there are no separate edges intersecting, these shapes have no vertices. A semi-circle also has no vertices, because the intersections on the semi-circle are between a curved line and a straight line, instead of two straight lines.
Vertices of Three-Dimensional Shapes
Vertices are also used to describe points in three-dimensional objects. Three-dimensional objects are composed of three different parts. Take a cube: each of its flat sides is called a face. Each line where two faces meet is called an edge. Each point where two or more edges meet is a vertex. A cube has six square faces, twelve straight edges, and eight vertices where three edges meet. In other words, each of the cube's corners is a vertex. As with two-dimensional objects, some three-dimensional objects -- such as spheres -- do not have any vertices because they do not have intersecting edges.
Vertex of a Parabola
Vertices are also used in algebra. A parabola is a graph of an equation that looks like a giant letter "U." The equations that produce parabolas are called quadratic equations, and are variations on the formula:
y = ax^2 + bx + c
A parabola has a single vertex -- either at the bottom point of the "U," if the parabola opens upwards -- or at the top point of the "U," if the parabola opens downwards, like an upside down "U." For instance, the bottom point of the graph of the equation y = x^2 is located at the point (0,0). The graph rises on either side of this point. So (0,0) is the vertex of the graph of y = x^2.