Android CM

CM is the abbreviation of CyanogenMod, which is a customized, aftermarket firmware for gadgets (smart phones, pads etc). CM is based on the open-source (means every developer can play with the source code following Generla Public License, many open source codes are even better than the commercial ones, famous ones are like Linux, Android, ZIP, FireFox,Eclipse, PHP,MySQL, Apache,...so many excellent ones).  You can understand CM as a firmware which can be used as a tool to root your devices allowing gain access to Google Android App Market.
      Current released version (relatively stable ones) are CM7 Alpha 3/3.5. The official CM website is http://www.cyanogenmod.com and the developer team is very active in xda developer forum http://forum.xda-developers.com/ and CM is actually a team work by CM developers and developers on XDA-Developers.
      CM7 is based on Android 2.3 Gingerbread and currently they are working toward to the next version CM9 which should be based on Android 4.x Ice Cream Sandwich.

Root, JailBreak and Legality

You may encounter many such jargons used by many who are in favor of up-to-date technology about "Gadget" (A gadget is a small technological object such as a device or an appliance that has a particular function typical examples are smart phone, tablet). 

Root: 

    Rooting enables users to use the superuser, or root, permissions of the Linux-based operating system (OS)(Android is based on Linux system) thus have more control over it,thus overcome limitations carriers or manufacturers put on your devices. One example is Amazon Kindle Fire, the Android on the Fire is extensively customized to allow users more easily buy stuffs from them, and has many limitations. Another example is the fire-saled HP TouchPad (TP), the WebOS does not have as many apps as on Android, thus a totally new Android has to be installed on the TP.

    Advantages of rooting are: more freedom to control your gadgets, fun to explore

    Disadvantages are: risk of "bricking" (Firmware cannot start and run on your device) your gadgets and loss your data; void your phone’s factory and/or service provider warranty  

JailBreak:         

   This is often applied to only Apple's system IOS. Hacking the iPhone, iPod touch and iPad allows users to