Beryl is, simply put, the best enhancement that desktop computing has seen in years.
Beryl is a combination composite and window manager. What all this mumbo-jumbo means is Beryl is a program that draws the borders of your program windows, and decides where and how these windows will appear.
This is better understood through experience.
This is tvtime, showing TV on my computer monitor. Note how hovering the mouse on the TV program's taskbar button creates a tooltip-like preview window. This effect really needs to be seen in video, because the tooltip is fully animated, with no delay -- whatever appears on the TV screen, appears on the tooltip as well.
The second screenshot shows the "desktop cube" from an angle. With Beryl, you get four "desktops" (screens), arranged as the sides of a cube, to place your windows. Using keyboard or mouse shortcuts, you can quickly switch to any of the four sides of the cube. What you're seeing right now is the manual drag using the mouse, showing two sides of the cube. During rotation of the cube, all four sides become semitransparent, so you can see all windows at a glance.
If you're using Linux, and you have a fairly recent computer, download and install Beryl using your favorite distribution's software management tool. You won't regret it.