In marine or naval terminology, the dog watch is the period of time (or work shift) between 1600 and 2000 (see ship's bells). It is split into two (first and second dog watches).
The name is thought to have come from the fact that someone tasked with one of these 'half' watches was said to be 'dodging the watch', taking or standing the 'dodge watch'. This became shorted to 'dog watch' (the joke that this watch is cur-tailed is a little dogged!).
An alternative etymology is that those sleeping get only 'dog sleep' in this watch.
The reason behind this watch's existence is that in order for the crew to rotate through all the watches it was necessary to split one of the watches in half. This allowed the sailors to stand different watches instead of one team being forced to stand the mid-watch every night.
The choice of time also allows both watches (if there are only two) to eat an evening meal at about the traditional time.