A center is a point that is the same distance from all the extremities of a figure.
The center of a circle is the point which is equidistant from all points on the circle. In the figure below, C is the center.
The center point is often used to label the whole circle. The figure below would be called "the circle C".
The center of a regular polygon is the point that is same distance from all its
vertices.
For an overview of four common triangle centers see
Overview of triangle centers.