A line shape function is a (mathematical) function that models the shape of a spectral line (the line shape aka spectral line shape aka line profile). The specific shape of the line i.e., the intensity at each wavelength along the width of the line, is determined by characteristics of the source and the medium. For example, one source of line broadening is Doppler broadening due to movement of particles emitting or absorbing, for which temperature is a factor. By mathematically modeling each such factor (each with its own characteristic line shape function), the shape of a line given a set of circumstances can be derived, and conversely, an observed line of that shape can reveal the characteristics, or if more than one set of characteristics could create the shape, then constraints on the characteristics.
A Voigt profile is a model line shape created by the combination of model line shapes of two broadening mechanisms: a Gaussian function for Doppler broadening and a Lorentzian function for pressure broadening. The combined line shape function is a convolution of mathematical functions for each, but a linear combination of the two can serve as an approximation. The Sobolev approximation consists of a technique to make line shape models tractable despite an additional complication.