André's comment is right; the parameter $\beta$ has a particular meaning in certain settings.
Here's an example. One common use of the gamma distribution is that if $T$ is the amount of time until the $n$th arrival in a Poisson process with rate $\lambda$, then the pdf of $T$ is
$f_T(t) = \frac{ t^{n-1}\lambda^n e^{-\lambda t}}{\Gamma(n)}, \ \ t> 0.$
(Poisson processes are often used to model arrivals and departures in queuing systems.)
Since $\lambda$ is a rate, it represents the average number of arrivals per unit time.
If we rewrite the pdf with $\beta = 1/\lambda$ we get the form in your question,
$f(t| n, \beta) = \frac{1}{\Gamma(n) \beta^n} t^{n-1}e^{-t/ \beta},\ \ t > 0.$ Given the interpretation of $\lambda$ above, $\beta$ represents the average amount of time between arrivals; i.e., the average interarrival time.