If you've ever used GNOME 2 in the past and missed it, this is the Linux desktop environment for you. New users may be better off going with Cinnamon, which is a traditional-style desktop built on more modern code. However, older computers may perform better with MATE 2. If you have issues with performance on Cinnamon and you want something similar, give MATE 2 a try.
Unlike more modern desktop environments, desktop effects requiring 3D acceleration aren't activated by default on MATE. But it's now become easy to enable Compiz and get those visual effects on a MATE desktop.
Use Linux Mint with MATE for a solid MATE experience.
Lxde currently takes the crown for lightweight-but-user-friendly desktop environments. If you have an older computer you want to keep in the game, try Lxde on it instead of modern Linux desktop environments that need more graphics hardware, CPU time, and RAM to run properly. You won't find fancy graphical effects here, but you get a basic, lightweight desktop environment that lets you launch and manage your applications.
Try Lubuntu for a solid Lxde experience.
Xfce is in a bit of a weird place right now. Traditionally, this was the primary third option after GNOME and KDE. It was a bit more lightweight than KDE and GNOME. After GNOME 3 was released, Xfce became the top "traditional" Linux desktop environment.
Since then, Xfce has been squeezed from both ends. Lxde is just more lightweight. Cinnamon is more full-featured. MATE is more fully featured and almost as lightweight at this point. Xfce isn't dead, but its development is moving slowly.
Give Xubuntu a spin if you'd like to try Xfce.
Mix and match
You can try another desktop environment by booting up a distro that comes with it. Or, install a different desktop environment on your current Linux system and switch between available desktop environments using the session option on the login screen. That kind of flexibility is part of the beauty of Linux.
Speaking of, these are just the most popular Linux desktop environments you'll hear about--there are many, many more out there.
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