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Full width at half maximum (FWHM) is one measure of the width of a hump or valley in the plot of a continuous function. (The Gaussian function with its bell curve is an example of a function with such a hump.) "Maximum" is taken as the maximum amount above (or below, i.e., "maximum depth") the value of the function on either side of the hump or valley, and the value is the width in the "x direction" between points on this hump or valley where the function is half the difference between this maximum (or maximum depth) and the value outside the hump/valley. Astrophysics uses the measure as a means to quantify the width of a spectral line, e.g., in formulae that relate the width to other quantities. It is also used to specify passband specifications, sensitivity functions and other similar functions.
A similar term, FWZI for full width at zero intensity is the width of the spectral line right at the continuum, typically used for broad lines.
The term half width at half maximum (HWHM) clearly means half the above FWHM. Regarding the un-modified term half-width, I have seen published definitions stating it is used for the HWHM and others saying it is used for the FWHM, as if it were short for "the half-maximum's width".