The Display Manager allows you to enter different environments and depending on the display manager you have, other options may be available, such as choosing your window manager, setting up display manager, etc..
Desktop Environments are your GUI generally to run your OS, although it is still a shell. So in Windows you have Windows Explorer, alternatively you could install KDE, and now your shell is KDE. So for DE on Linux you have multiple options, such as Gnome, Unity, Mate, LXDE, Openbox, KDE, the list goes on and on. While I have multiple environment sets, it is not particularly ideal if you find yourself saying things like "The future of Linux is Mint!" Maybe it is, maybe it isn't. You will have troubles experimenting because you will have to learn first that you are pulling either Ubuntu or Debian Repos, so you must change to "The future of Linux is Mint running on Ubuntu and Debian!", and your journey should be easier.
Window Managers control your windows within the environment(shell). Some DE only run on their own window manager, like Cinnamon, which uses Muffin. Compiz, theoretically can run on Cinnamon, but not very well, as it is more based on Gnome(3), otherwise known as the DE that no one uses. Gnome 2 is known for being able to adapt to almost any Window manager, which is cool, but has misled people into believing that this is true with every DE. IT IS NOT. Gnome 2 is special. It is also retired, so this is why you are seeing Mate on Ultimate Edition 3.4 lite. This is why I feel like screaming when I see Compiz/KDE, because it is pointless. Would you take a Honda Engine and put it in a Chevy Truck? No. The Honda engine sure, can work with other small vehicles, but will be of little use in the truck that has more weight, inevatably, costing you just as much gas and much pain and sorrow. Vice Versa, the same is true, except a Honda with a big V8 is probably pretty fun, either way, both vehicles will be prone to breaking down.
None of this is hard and fast, I am just trying to alleviate some of the confusion without getting extremely technical, and there is much technical to this.