The gamma function is defined with two parameters, alpha, and beta, over the +ve non-negative real line. alpha can be any real number greater than 1 unlike the Poisson likelihood where the equivalent quantity are integers (values less than 1 are possible, but the function ceases to be integrable) and beta is any number greater than 0.

The mean is alpha/beta and the variance is alpha/beta². Conversely, given a sample whose mean and variance are known, one can estimate alpha and beta to describe that sample with this function.

This is reminiscent of the Poisson distribution where alpha ~ number of counts and beta is akin to the collecting area or the exposure time. For this reason, a popular non-informative prior to use with the Poisson likelihood is gamma(alpha=1,beta=0), which is like saying “we expect to detect 0 counts in 0 time”. (Which, btw, is not the same as saying we detect 0 counts in an observation.) [Edit: see Tom Loredo's comments below for more on this.] Surprisingly, you can get less informative that even that, but that’s a discussion for another time.

Because it is the conjugate prior to the Poisson, it is also a useful choice to use as an informative prior. It makes derivations of formulae that much easier, though one has to be careful about using it blindly in real world applications, as the presence of background can muck up the pristine Poissonness of the prior (as we discovered while applying BEHR to Chandra Level3 products).