This article is from the Amiga FAQ, by Ignaz Kellerer with numerous contributions by others.
MUI is a set of shared libraries which are used to build a comfortable GUI (`Graphical user interface'). The general idea of MUI is that the programmer determines only the logical structure of the GUI and the look (Font, Size, Windows on Workbench, Public Screen, own Screen, ...) is determined by the user only. From the programmers point of view using MUI is simpler than `gadtools.library' but much more powerful. On the other hand MUI interfaces are slower than GadTools-Interfaces, especially on old 68000 machines.
MUI is found in two archives, one for developers only and one for users. Source: Aminet, `dev/misc'.