Milli-Q Water is simply water purified using a Millipore Milli-Q lab water system. Over the past 30 years there have been several different Milli-Q water systems and each can have up to 4 different configurations.
The term Milli-Q water usually comes from a procedure written in a laboratory that had a Milli-Q water system and instead of calling for Type 1 water or some other water specification, they simply called it based on the equipment available in the lab of whom ever wrote the procedure to begin with. The basic Milli-Q produced water meeting NCCLS, CAP or ATSM Type I water, having 18.2 megohm ionic purity with little if any bacteria.
Other versions can also give ultra low total organic contamination (UV) and Pyrogen free water (UF)or a combination of all of the above (UV/UF).
You will also find the same thing called 'Nanopure Water' or 'DI Water' but all of these terms are vague when discussing the actual water quality needed for a given procedure.