are older terms for MUSLIM and ISLAM used in many publications up to the middle of the twentieth century. Occasionally Muslims are upset about these terms because of the wrong impression that Mohammedan means "somebody who worships Mohammed" as a Christian is "somebody who worships Christ".
This is however a misunderstanding, and just as the word Christian means "one who is a follower of the Christ", so did the word Mohammedan signify a person who follows the message and example of Mohammed.
Muhammad himself addressed the Muslims (sometimes) as "followers of Muhammad":
Narrated 'Aisha : ... The Prophet then said, "O followers of Muhammad! By Allah! There is none who has more ghaira (self-respect) than Allah as He has forbidden that His slaves, male or female commit adultery. O followers of Muhammad! By Allah! If you knew that which I know you would laugh little and weep much." (Sahih Bukhari 2.154)
The term is actually even used by Muslims. Hughes' Dictionary of Islam, published 1886, states:
MUHAMMADAN. Arabic Muhammadi. A name seldom used in Muhammadan works for the followers of Muhammad, who call themselves either Mu'mins, Muslims, or Musalmans. It is, however, sometimes used in Indian papers and other popular publications, and it is not, as many European scholars suppose, an offensive term to Muslims.
This is even confirmed by a recent Muslim (online) publication, elaborating on "the greatness of al-Haqiqa al-Muhammadiyya, the ‘Muhammadan Reality’" (Source).
Dictionary entries from the Hypertext Webster Gateway:
Mohammedan \Mo*ham"med*an\, a. [From Mohammed, fr. Ar. muh['a]mmad praiseworthy, highly praised.] Of or pertaining to Mohammed, or the religion and institutions founded by Mohammed. [Written also {Mahometan}, {Mahomedan}, {Muhammadan}, etc.]
From WordNet (r) 1.6 (wn)
Muhammadan adj : of or relating to the Arabian prophet Muhammad or to the religion he founded [syn: {Muhammadan}, {Mohammedan}] n : a believer or follower of Islam [syn: {Muslim}, {Moslem}, {Mohammedan}, {Muhammedan}, {Muhammadan}]
Although there is nothing disparaging or wrong in those terms, and they were not ment to be insulting, they are outdated. It should be a matter of common courtesy to use the words Muslim for the believer and Islam for the religion since those are the terms used by the Muslims themselves.
However, there are situations in which those terms are still useful. Though the dictionary meaning of "Islam" and "Muslim" are simply "submission" and "one who submits", many Muslims when asked for the meaning of these words reply with "submission to God" and "one who is in submission to God". However, Jews and Christians also understand themselves as being in submission to God (in obedience to those scriptures that they accept as God's true revelation). Therefore, they do not want to accept the term "non-Muslim" for themselves with its obvious connotations as refering to those who are not in submission to God.
If some religous group decides to call itself "the honest" what would all those be who do not agree with their teaching and oppose this group? The choice of names can be a clever trick to put oneself into a position of having a superior claim to integrity and "being right" even before the claims and teachings of the religion in question are evaluated for their truth.
For this reason some prefer "Mohammedan" as a more neutral term for the purpose of refering to those who follow the life and teachings of Muhammad, leaving open whether they are from God or not. However, those who want to use it, should carefully explain the meaning and their reason for the choice of this term because of the common misunderstanding as insult which term carries with many Muslims.
See also this more detailed discussion of the term MUHAMMADANISM.