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A Gaussian function is a function of this form:
f(x) = a e(-(x-b)²/2c²)
for constants a, b, c, and Euler's number e, 2.71828... The adjective Gaussian means "described by this function". The function plots a bell curve, and among its uses is that it is the probability density function of normal distributions (i.e., Gaussian distributions), which, as per the central limit theorem, describe probability distributions of summing a specific number of independent random variables. For example, errors in measurement are commonly assumed to include a purely random element and modeled using a normal distribution and thus a Gaussian function. When representing a normal distribution, a (per above) has this relation with σ, the standard deviation:
a = 1 / ( σ √(2π) )
Some sources of line broadening show a Gaussian profile, i.e., they are described by a Gaussian function.