Common name for distributions that are bounded on one side Is there a common name to refer to distributions that are bounded on one side, and unbounded on the other side? For example, log-normal distribution, where the minimum value is zero, the maximum is +infinity.
 A: I would say no, as there are quite a few distributions in addition to log-normal that have support on $[ 0, \infty)$, such as the $\chi^2$ or gamma. (Wikipedia even has a list devoted to this criterion.)
In practice there are different circumstances in which such distributions are useful to approximate actually observed and measured phenomena. For one example, in some situations you may use the term censored at the boundary to describe how observations can go above and below the boundary, but they can only be measured within a certain support (and when they go outside this support they are recorded as being at the end point and/or beyond). This is typically referred to when you have an instrument that can not measure the numerical value outside of the bounds, it only knows it is at or below the boundary. For log-normal an example is the measurement of $\log(\text{wages})$. I believe for the US census they censor the distribution at $0$, although people who own their own businesses can be in the negative. (Some macro economic variables are well approximated by a log-normal distribution, but the support of the actual micro level units is partly in the negative.)
Another example (as user41315 mentioned) are truncated distributions. Truncated means "chopped off". Sometimes we only observe/record the measurement if it exceeds the boundary. For another economic example of wages, lets say you only had to file taxes if your wages were above $0$. So it is not like the census that just records $0$ even if you have less than $0$, you just simply don't observe the individuals with less than $0$ wages. As whuber stated in the comment, you can take take any distribution and re-express it as a truncated one.
The descriptions of censored or truncated refer to how the data are measured, and not to particular distributions. Not all measurements that are bounded on $[ 0, \infty)$ are necessarily truncated or censored though. For example distances or squares of values we know can not go below $0$.
A: There seems to be no standard term, based on my experience.  Some people refer to them as "one-sided", or "of one-sided support".  I'd say "supported on [the positive half-line]", even though that is clunky.
Edit: related to Andy W's comment- "truncated" rather than "censored" more appropriate if the values outside a certain range are ignored (without adding mass to the observation window's endpoints).  [I don't have enough reputation to comment directly on your answer]. 
