A parabola has one vertex: it is the point of greatest curvature, and it is usually considered to be the centre of the parabola. For graphs of the form y = (ax - b)2 + c, the vertex is at (b/a, c). If a is positive, this is the lowest point on the graph, if a is negative, this is the highest point.
The vertex of a parabola is the only point which is both on the parabola and on the parabola's axis of symmetry.
A hyperbola has two vertices. Again, these are the points of maximum curvature, and they also lie on an axis of symmetry (hyperbolae have two axes of symmetry, but one of them does not touch either curve). They are the two points, one on each 'side' of the hyperbola, which are closest to each other.
Stretching the definition, the two points of greatest curvature on an ellipse may also be considered vertices. These points are on the major axis, the two points on the ellipse which are farthest from each other. This is a rare usage of the word, however.