Moe is a traditional English nickname; it has become so familiar that it is no longer as popular as it once was. It has strong working-class overtones, and because one of the popular Three Stooges was named Moe, it also has overtones of dopiness.
Moe is short for four unrelated names:
Elmo, which is in turn a shortened form of two unrelated names; Erasmus (from Greek elene meaning 'torch'), and Aldelmo (from Proto-Germanic helmo, meaning 'helmet').
Morris, which comes from the Latin Maurus, meaning Mauritanian. This is the same root as gave us the word Moor. Other forms of Morris include Maurice and Morey.
Moses from the Hebrew Mosheh. The origin of Moses is not known, but it is obviously a popular name due to the whole 'writing the Torah' thing.
Muhammad from the Arabic Mahum or Mahimet, meaning 'Praiseworthy'. This is another name made famous by a holy prophet, in this case the writer of the Quran.
The name Moe was somewhat popular given name (as opposed to nickname) in America around the turn of the last century, popping into the top 1000 boys names from 1890 to 1920. It has not entered the top 1000 since then, and it is not likely to do so in the foreseeable future.
Moe is also a surname originating in Norway and Sweden, where it also occurs in the forms Mo and Moen. This form comes from Old Norse mór, which came to mean ‘plain' or ‘moor’ in Norwegian and 'sandy heathland' in in Swedish.