The moniker is the name of any creature's genome file. In Creatures and Creatures 2 this was a 4 character alphanumeric string - automatically allocated monikers were in the format numberletterletterletter, eg. "1VWY", but monikers like "lucy" or "eve_" were also valid.
In later games using the Creatures Evolution Engine (from Creatures 3 onwards) the moniker-generating mechanism became more complicated; the format for these gives monikers like "001-norn-xvjbp-xyhye-ign3a-drjyp". The first number is the generation number (the higher of the parent's generations, plus one), the second indicates the species, and the rest are constructed from the genome. This increased length decreases the likelihood of identical monikers.
Importing two creatures with the same moniker can cause the game to crash, which is why it is not a good idea to try to export a creature and then import it twice.