In laymen's language, when no application is open, you are interacting with gnome-shell. When you are going to change system settings, you are interacting with gnome shell. When any program is running, it requires a GUI. Gnome provides this GUI using a toolkit known as GTK. Gnome 2 used GTK2+, Gnome 3 uses Gtk3+. Gnome also has their own set of default application which are designed to work with Gnome 3. Please correct me if I am wrong.
Gnome 2.32 was the most popular Linux Desktop Environment (DE).
Things were going all well, then something unexpected happened. UNITY was introduced to Desktop. Earlier it was only available for Netbooks. it did not offered customization of DE and was not accepted well. Then came another 'unholy mess'
Gnome 3 (with gnome-shell).
New version did not offer customization. Critics said it was not optimized for desktop PC, but for touch screen devices. You could not have icons on desktop and cannot configure it the way you like.
When someone says Gnome3, people take it for granted that it is gnome3+gnome-shell.
According to some users, Gnome2 was very mature and there was no need to change it. So Gnome2 was forked into MATE. MATE uses GTK2+ like gnome 2 and tries to behave exactly as gnome2. To be able to install MATE, the lib were conflicting. As Vincent has said, MATE Dev Perberos (MATE dev)had to rename anything tagged 'gnome' to avoid conflict. So MATE is like a replacement for Gnome 2. It is a completely new DE.
To make Gnome3+gnome-shell usable, Clem created Mint Gnome Shell Extension (MGSE). These are just 3 extensions, that try to make Gnome3 more desktop friendly. But they were not enough and Gnome 3 devs were not interested in optimizing for desktop PC. So clem had to fork gnome-shell and create Cinnamon.
Like UNITY, Cinnamon runs on top of for in simple words along with Gnome3. comice OS also has forked Gnome-shell to suit their distro.
KDE 4.6 is also designed keeping in mind touch screens. So there was a drought in DE. people shifted to XFCE / LXDE or tried to adjust with Gnome-shell and KDE. In mean time, Cinnamon was developed and is highly appreciated by Linux Community in general.
Hope this helps.EDIT: sorry for double posting. Deleted last post.
EDIT2: Gnome-shell linkshttp://live.gnome.org/GnomeShellhttp://live.gnome.org/GnomeShell/FAQ#Wh ... E_shell.3F