A WindowManager is a program that, well, manages your windows. Without it, you wouldn't be able to move or resize windows on an X11 server. The WindowManager is responsible for the borders around windows, their titlebar, and so on, and is the program that reacts to actions in these areas, allowing you to resize it as well as minimize, maximise, or close windows. Some WindowManagers can be run standalone, others (such as XFce's Xfwm, KDE's KWin, and GNOME's SawFish/Metacity) are intended to be used as part of a DesktopEnvironment.

A list of pages about WindowManagers in this wiki:

The Windowmanagers for X page lists and describes many more.