The phrase “Peace Be Upon Him” is used by Muslims as a mark of respect and veneration for Muslim prophets and other figures of respect. The use of this honorific is mentioned specifically in the Qu'ran, and also in other supportive texts used by Muslims, such as the Hadith. Non-Muslims are not expected to say this phrase when discussing Muslim prophets, although they certainly may do so.
Use of this honorific is known as salawat, or as saying salah. Salah means “prayer,” and all devout Muslims are expected to integrate daily prayer into their religious practice. By saying salawat for someone, Muslims pray for the individual under discussion. You may see salawat abbreviated in text as PBUH or SAW, although some Muslims frown upon this practice when the prophet Muhammad (Peace Be Upon Him) is under discussion.
According to the Qu'ran, God says salah for the prophet, and asks the angels to do likewise. The Qu'ran goes on to importune pious Muslims to say salah for the prophet when they see him and discuss him. The prophet Muhammad (Peace Be Upon Him) himself adds that Muslims should always say salah for him, and that by praying for the prophet, Muslims will be blessed. Each time the salawat is said, it is supposed to lift some of the sins of the person who says it, and some Muslims believe that the more times someone says salawat in life, the closer he or she will be to the prophet in heaven.
A number of figures are recognized as prophets by Muslims, including many Biblical prophets. The formula of the salawat is the same for all of them, although some Muslims believe that it is acceptable to abbreviate the salawat for some of these prophets. Islamic scholars have yet to rule definitively on the issue of abbreviating salah, although evidence in the Hadith, seems to support the idea that the salah should not be abbreviated.
Like many religious honorifics, the use of “Peace Be Upon Him” is intended to remind people of a prophet's special place in the practice of Islam, and to encourage veneration for important figures in Islam. Some people also say salawat for the dead as a mark of respect, especially when talking to their family members.