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A Mach number is a ratio of the relative velocity of a flow of a fluid compared to something else versus the fluid's speed of sound, i.e., a Mach number of 0.5 means the relative flow velocity is half the fluid's sound speed. The "something else" could be an object that the fluid is passing or striking, or it could be other fluid (with a different flow). The Mach number's general range is a useful indicator of a regime of fluid flow characteristics, regimes such as subsonic, supersonic, and transonic (i.e., with a Mach number close to 1 when some of the fluid flow is likely to be subsonic and other to be supersonic). Each of these regimes (and in various sub-regimes within these) is associated with certain kinds of turbulence and/or shock, and certain approximations of fluid dynamics are effective, for example, for the design of aircraft.
An Alfvén Mach number is such a ratio with the speed of Alfvén waves in a plasma (rather than with its sound speed). The term sonic Mach number is sometimes used for the ordinary Mach number when both types are in discussion. A use of the Alfvén Mach number is modeling and characterizing solar weather.
Both these Mach numbers are used in the study of the dynamics of molecular clouds and dense cores, to characterize turbulence according to the relative velocities of adjacent fluids. They are of interest in determining the effect of turbulence, jets, and winds on the stability of parcels of high-density gas, essentially to study the particulars of star formation feedback.