There are probably many probability distributions that have just one parameter. The most important one for statistical analysis is probably the Student t distribution.

This probability distribution is fully described by a single parameter which is often called "degrees of freedom". The parameter describes the scale of the distribution, and not the location, since the Student t distribution is always centered at zero (unlike the normal distribution, which has a scale parameter, the variance, and a location parameter, the mean).

Another example of a distribution that is described with a single parameter is the exponential distribution. Unlike the Student t distribution, it is a distribution that takes only positive values.