The normal distribution is often used to model phenomena that give strictly positive results. Example: Heights of women in a particular population with mean $\mu = 67$ inches and standard deviation (SD) $\sigma = 3.5.$
Technically, using a normal is 'wrong' because a normal distribution has a left
tail that extends to negative infinity, and obviously no women have
negative heights. However, almost all probability is contained within 3 or 4 SD of the mean, so zero (almost 20 SD below) plays no practical role.
Similarly, SAT (and other) exam scores are sometimes described as normal, totally ignoring the minuscule probability that would correspond to nonexistent negative scores.
The gamma family of distributions places all its probability on
the positive half-line. Gamma distributions are right-skewed with
mode < median < mean. The shape is governed by a shape parameter.
For larger shape parameters
the distribution has a left sided tail and a somewhat more pronounced
right sided tail. See graphs of various gamma densities in the Wikipedia article. (A scale or rate parameter helps to determine the
variance of a gamma distribution.)
Gamma distributions are used to model waiting times, and many other phenomena in social, biological, and physical sciences. The gamma family of distributions is not as widely used as the normal family, but if
any family of continuous distributions can be described as the 'default' non-negative family, the gamma family would be the prime candidate.
In queueing theory (concerned with the
behavior of waiting lines), gamma distributions with integer
shape parameters are called Erlang distributions. The chi-squared family is another sub-family of gamma.
Exponential distributions are members of the gamma family with shape parameter 1; strongly skewed with no left-sided tail.
A few other particularly useful continuous non-negative families, among many, are the lognormal (taking logs of data
produces normality), Weibull, Rayleigh and Pareto. (Also discussed in Wikipedia articles.)