If you already know the diameter endpoints, the center is the middle node(s) of that path. Run BFS from one endpoint, track parent pointers, and reconstruct the path to the other endpoint.
Walk halfway along the path to find the center. If the diameter has odd length, there are two centers (the middle edge's endpoints). If even length, there is one center node.
The center minimizes the maximum distance to any other node. This property is useful in facility location problems and network design.