A common example is NMR. In this process, the nuclear spin of an atom starts rotating, with the frequency of rotation proportional to the external magnetic field that the atom experiences. However, in an inhomogeneous medium, the magnetic field often varies from point to point (depending, for example, on the magnetic susceptibility of nearby atoms), so the frequency of nuclear spin rotation is different in different places. Therefore, when detecting the resonant rotation frequency, there is a linewidth (i.e., finite range of different frequencies) due to the variation in that resonant frequency from point to point. (This is called "inhomogeneous broadening".)
However, if the atoms are diffusing around the system, they will experience a higher magnetic field than average sometimes, and a lower magnetic field than average other times. Therefore, (in accordance with the central limit theorem), the time-averaged magnetic field experienced by an atom has less variation than the instantaneous magnetic field does. As a consequence, when detecting the resonant rotation frequency, the linewidth is smaller (narrower) than it would be if the atoms were stationary. This is the motional narrowing effect.