How does it work

Linux has a completely different philosophy about graphical desktops.
Variety is the Theme!
The graphical desktop can be split into several parts: the X server, the X clients, the windowmanager and, recently, the "desktop" (but this is something different to a M$Win desktop).

The X server is hosted by the PC (or graphical terminal station) you are sitting in front of. An X server is nothing more than the black-white-chess pattern you see very shortly before the graphical background of your windowmanager appears. The X server handles the graphical presentation and is responsible for the communication between the hardware (in particular the graphic card) and software (the X programs).
If you only ran an X server, no work would be possible.
It offers neither a menu, nor windows or any other features you need.

The work on the X server is done with X clients. These X clients use the libraries that are integrated in the X server to display graphics and graphical frontends.
If you see an X term on your monitor, it is an X client.
The communication between X server and client works through the network. That is the reason why you can start an X client on any computer in the network and see it somewhere else (this is the way the X terminals work; you start the programs on a powerful server and sit in front of a simple terminal).

Even if you do not own a network card, the communication works via network. It uses the dummy device Loopback.

X server and X clients are not the whole world.
Useful functions like "Maximizing", "Minimizing" and "close window" are not included in the functionality of server and clients but brought to you by a window manager.

A long time ago, just three of these components existed. However, for the past three years, an additional "thing" has existed: the desktop.
The desktop offers similar functionalities as M$Win: put icons and files on the background and start these with a double click.
Of course, icons existed long before (e.g. after minimizing a window with FVWM), but the functionality was not as great as with M$Win.
If you wanted to start programs via small icons, you would have to use the button bar (which you could compare with the M$Win start bar).
KDE and GNOME are two popular desktops running on Linux.
They come with their own windowmanagers, but both programs could be used with other windowmanagers to use alternate options as well.

Oldies, but Goldies

A lot of "old" windowmanagers exist which are still in use, in particular on older computers, because they do not need the newest hardware and also run quite fast on small terminals.

One of the very first windowmanagers for Linux was FVWM (The windowmanager is so old that no one knows what this abbreviation stands for). The next generation of FVWM offers a lot of features like a button bar and the virtual desktop presenter. The configuration is done in the file .fvwm2rc which is very long and has a special structure. The configuration is often overwhelming because it uses a special script language. Homepage: www.fvwm.org

MWM

MWM is a windowmanager using Motif. This commercial widget set can now be replaced by lesstiff, a free and powerful substitute that works without any problems in combination with MWM. Motif is known as a library that needs a lot or resources and is hard to configure. Maybe that is the reason why the MWM windowmanager has not become very common. Homepage:www.lesstif.org

TWM

TWM is a minimalistic windowmanager that just offers the basic functions. It is used on slow computers or in case of an emergency. If you use VNC to access another Linux computer, it is used for the default windowmanager. Homepage:http://linux.about.com/compute/linux/library/bltwm.htm

CDE

CDE is a commercial desktop that is mainly used on commercial Unix clones. It was assumed this windowmanager to be standard, but the Common Desktop Environment has never reached a wide acceptance. Homepage:http://www.opennc.org/tech/desktop/cde/

Qvwm/fvwm95

These two windowmanagers try to offer a M$Win look-and-feel, but fall short. Both are recommended if you change from M$Win to Linux to make the first few days easier. Homepage: www.qvwm.org Homepage: ftp://mitac11.uia.ac.be/html-test/fvwm95.html

Afterstep

Afterstep is the Linux version of the popular NextStep framework that is in use on many commercial Unix clones. Afterstep is a very colorful windowmanager with its own button bar (Wharf). Many programs exist that can be included into the Wharf and show e.g. CPU, network, RAM and other system information. Afterstep has a big resonance in the Unix community but it seems that its successor, WindowMaker, is becoming more and more popular. Homepage: http://www.afterstep.org

Newcomer

WindowMaker

WindowMaker wants to be the successor of Afterstep, in particular in the visual presentation. It is not based on Afterstep but a complete new program. The motto "make it smaller, faster and with more colors" seems to be followed during the development. It offers a lot of optical finesse without a slowdown of the machine. The presentation can be adjusted to your own wishes but keeps the classical NextStep design. It can be used on slow computers although it is so colorful. Similar to the Wharf of Afterstep many DockApps exist for WindowMaker, too. (also see e.g. Issue 2: WindowMaker and Gnome) Whoever wants to change the look of WindowMaker will find a lot of themes at wm.themes.org. The support of Gnome and/or KDE program support has a special priority in the development. Homepage: http://www.windowmaker.org

Enlightenment

Enlightenment is the recommended windowmanager for all fanatic designers. This windowmanager does not care about speed or a compact, efficient program but tries to use all the features a graphic card has. No other windowmanager offers as many graphical features. It is only recommended on a PC faster than a PII with a 8MB Ram graphic card. Meanwhile there is also a big collection of themes for Enlightenment (available at e.themes.org) and you can change the look so much that you do not recognize that it is Enlightenment. Themes exist that give Enlightenment the look and feel of BeOS, MacOS and also a StarTrek console. Sometimes it is hard to guess that the same windowmanager is working in the background. Not all themes try to change the look and feel; some let the desktop appear like art. Also for Enlightenment, programs exist that show information of the system in a nice graphical way; these are called Epplets. Homepage: http://www.enlightenment.org/

tiny helpers

This list is far from complete. There exist a lot of more windowmanagers which are usually designed for a special group of users.
Here are some of them:

amiwm

AmiWM is a clone of the Amiga desktop (Amiga Workbench 3.x). This spartan and efficient (but nowadays dusty) desktop can be a nice place for Amiga fans. Too bad that we did not find any nice tools and system applets for AmiWM. Homepage: http://www.lysator.liu.se/~marcus/amiwm.html

Sapphire

Sapphire is a small windowmanager which works quite fast on slow computers with nice small features. They changed the programming language from C to C++ that is not common in the Linux community. Sapphire is a further development of Aewm (Aesthetic Window Manager) but also owns parts of the Blackbox windowmanager. Homepage:sapphire.sourceforge.net/

Sawmill

Sawmill can be configured infinitely. All window decorations and user interfaces can be configured using the script language Lisp. It concentrates on the presentation of windows and menus. Icons, backgrounds and special applets are not included into Sawmill. Meanwhile there is a GTK configuration program that makes the adjustment to your wishes much easier. (Update: Sawmill got the new name - Sawfish - and is set to be the default windowmanager in combination with Gnome (used to be Enlightenment as the default) in many distributions.) Homepage:http://www.dcs.warwick.ac.uk/~john/sw/sawmill/

Twin

Twin is a windowmanager completely different to all others. Twin is a windowmanager for the text console and can be comfortably used with a mouse. Windows and menus are presented with ANSI characters, but you cannot start every X program. Just special versions that are adjusted to Twin (many of these exist e.g. a CPU display and CD player). It does not need big graphic cards but also it does not really replace a more traditional windowmanager. Homepage: http://linuz.sns.it/~max/twin/

Blackbox

Blackbox has one aim: speed. Nevertheless, the windowmanager looks great. KDE users might prefer Blackbox because it works very well with KDE programs; support for Gnome has not been implemented Homepage: http://blackbox.alug.org/

XFCE3

XFCE3 is based on GTK+ and has a nice presentation. In addition, it offers a few applications that support the work with the desktop. Next to a panel, file manager and background manager many other programs are included. Homepage: http://www.xfce.org/

IceWM

IceWM is a windowmanager that is designed to be fast. The default setup makes sense and you do not need to configure much. In addition, the windowmanager can be used without a mouse and offers a support of several themes: "ultracute" is quite nice. Homepage: icewm.sourceforge.net