LMDE is a Linux Mint distribution, containing all of the nice Mint tools (installer, updater, menus and other utilities), plus lots of preloaded and preconfigured packages, but built on top of Debian GNU/Linux, rather than Ubuntu as the numbered/named Mint distributions (currently 14/Nadia).
The pros and cons of Debian
Specifically, it's built on the Testing Branch of Debian — currently the 'Wheezy' release (although probably not for much longer). There are pros and cons to this - if you don't like Ubuntu, or you really prefer to start from a cleaner, more basic distribution, then being based on Debian is a big plus.
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