In English texts written by Muslims, the acronym PBUH stands for Peace Be Upon Him. After making a reference to Muhammad, Muslims say "Peace be upon him" in order to bless and honor Muhammad; see also the abbreviation SAW for the corresponding Arabic expression. [It is not very common, but occasionally one also see the acronym PPBUH (prayer and peace be upon him) which corresponds to the below discussed expression alaihi al-salat wa al-salam.]
Muslims are commanded to add this blessing after mentioning Muhammad's name or his title in speaking or writing. This is based on the quranic injunction in al-Ahzab 33:56:
The above is Pickthall's rendering of S. 33:56, and most Muslim translations are similar. However, this translation is not entirely accurate and hides some problems associated with this verse. For a detailed discussion, see the article The "Mystery" of PBUH revealed.
Exclusive to Muhammad
A noteworthy observation is that in English Muslims often speak the phrase "peace be upon him" or put (PBUH) into a written text after mentioning any one of prophets, thus creating the appearance of giving equal honor to all of them. The original Arabic expression Salla Allahu alaihi wa sallam is, however, used exclusively for Muhammad. All other prophets receive honoring phrases of a lesser kind.
After making reference to any of the other prophets Muslims say alaihi al-salam which means "upon him (be) the peace"; written form in English texts: e.g. Ibrahim (AS).
After mentioning the name of Muhammad Muslims either say the above mentioned phrase Salla Allahu alaihi wa sallam or alternatively alaihi al-salat wa al-salam which means "upon him (be) the prayer and the peace". In either case, Muslims ask for something more special and additional for Muhammad, i.e. "al-salat" (the prayer), and by doing so express that Muhammad is of higher rank than all other prophets.
Though we are not aware of an explicit prohibition in Islamic law to do so, most Muslims would look somewhat bewildered and perhaps even protest if anyone were to use Salla Allahu alaihi wa sallam or alaihi al-salat wa al-salam after mentioning Noah, Abraham, David, or any of the other prophets.
Exclusive to the prophets
Even this lesser phrase, alaihi al-salam, is not a general expression of honoring anyone that may be worthy of honor in the eyes of the speaker. Muslims would never use this phrase for a living or a deceased president of a country, even if he had been a pious Muslim. This phrase is used exclusively for the recognized prophets of Islam (at least by the vast majority of Sunni Muslims).
Therefore, speaking this phrase or writing the acronym PBUH after the name of anyone implies that this person is recognized as a genuine prophet of God, i.e. it becomes an implicit confession of faith in this person and his message.
Should Christians use the expression "peace be upon him"?
Sometimes Muslims complain that Christians do not speak with enough respect of the prophets because they do not say "peace be upon him" after mentioning the name of a prophet. There are mainly two persons in regard to whom this is an issue, Jesus and Muhammad.
Since Christians believe that Jesus is God Incarnate, i.e. the Lord God who came to this earth by becoming man, it would be an insult to add "peace be upon him" after the name of Jesus who is the GIVER — not the recipient — of all peace. It would be just as inappropriate for a Christian to to ask for peace upon Jesus, as it would be for a Muslim to say or write "... Allah (pbuh)". Doing so, would be giving Jesus explicitly less honor than he deserves, i.e. it would be dishonoring him publically.
Should Christians say "peace be upon him" after mentioning Muhammad?
Since Muhammad's message clearly contradicts and attacks the message of God's true prophets as recorded in the Bible, Muhammad cannot be accepted as a prophet of the Biblical God. Christians should neither give unnecessary offense when speaking about Muhammad, nor should they give the impression that they recognize his claim to prophethood. Using any of the Islamic phrases that are exclusive to Muhammad as the highest and final prophet of Allah or to the prophets accepted in Islam would be an implicit acknowledgement of Muhammad's prophethood and should therefore be avoided for the sake of clarity. See also our more detailed response to a Muslim who requested that we do so: Should Christians use the formula "PBUH"?
Other recipients of honorific phrases
After mentioning Ali, Abu Bakr, or any one of the other companions of Muhammad, Sunni Muslims usually say radi Allah anho (Allah be pleased with him), commonly abbreviated as RA.
A difference in Shia Islam
In addition to the prophets mentioned in the Qur'an, Shia Muslims commonly say alaihi al-salam (peace be upon him) also after mentioning the name of Ali, Fatima, al-Hasan, al-Husain and other members of Muhammad's household, particularly after the mentioning any of the twelve Imams of Shia Islam.
The following quotation is taken from a Shia glossary of Islamic terms:
A.S. refers to 'Alayhis-salaam, (God's) peace be with him. It is said after the names of all previous prophets, their mothers (e.g. Bibi Maryam [Mary] A.S.), the twelve divine Imams from the household of Prophet Muhammad (S.A.W.) and Bibi Fatima (A.S.). It will change to 'Alayhas-salaam (peace be with her) if it follows a lady's name. After any two names we say 'Alayhimas-salaam and after more than two names or when referring a group of people we say 'Alayhimus-salaam. Thus, we say Imams Hasan and Husayn (A.S.='alayhimas-salaam) and the Ahul-Bayt (A.S.'alayhimus-salaam). {Source}