Greeting (The salutation of peace to the Muslim when greeting and departing)

It is a recommended sunnah (way, tradition) to greet the muslim; it is
fard (obligatory) to reply.
Exceptions are that women are not expected to have to speak to non-mahram men.
The generally expected greeting is "As sala'amu alaikum" (peace be
uponyou) and the generally expected reply is"walaikum as sala'am" (and
unto you also, peace).
Adding "wa rahmatullahi" (and mercy) and/or "wa barakatuhu" (and
blessings) is a commendable act. If someone adds mercy and/or
blessings in their salaams to you, you should reply with the same or
add more goodness to it (i.e. if someone says "As sala'amu alaikum wa
rahmatullahi", you should reply with "As sala'amu alaikum wa
rahmatullahi" or also add"wa barakatuhu".
The Order to Greet
The Noble Qur'an An-Nur 24:61
...when you enter the houses, greet one another with a greeting from
Allâh (i.e. say: As-Salâmu 'Alaikum - peace be on you) blessedand
good. Thus Allâh makes clear the Ayât (these Verses or your religious
symbols and signs, etc.) to you that you may understand.
The Noble Qur'an An-Nisa4:86
When you are greeted with a greeting, greet in return with what is
better than it, or (at least) return it equally. Certainly, Allâh is
Ever a Careful Account Taker of all things.
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Ibn Katheer raheemahullaah explained this verse by saying: "If the
person greeted you with the Salaam you should reply in a better form,
or reply similarly. However, the extended form is preferable, but the
shorter is compulsory. Which means that greeting is preferable but the
greeting in return is compulsory, and it should be in the form of the
Salaam and not in any other form."
Hadith - Sahih Bukhari 4.543, Narrated Abu Huraira, r.a.
The Prophet said, "Allah created Adam, making him 60 cubits tall. When
He created him, He said to him, "Go and greet that group of angels,
and listen to their reply, for it will be your greeting (salutation)
and the greeting (salutation) of your offspring." So, Adamsaid (to the
angels), As-Salamu Alaikum (i.e. Peace be upon you). The angels said,
" As-salamu Alaika wa Rahmatu-l-lahi " (i.e. Peace and Allah's Mercy
be upon you). Thusthe angels added to Adam's salutation the
expression, 'Wa Rahmatu-l-lahi.' Any person who will enter Paradise
will resemble Adam (in appearance and figure). People have been
decreasing in stature since Adam's creation.
The Noble Qur'an Al-An'am 6:54
When those who believe in Our Ayât (proofs, evidences, verses,
lessons, signs, revelations, etc.) come to you, say: "Salâmun
'Alaikum" (peace be on you); your Lord has written Mercy for Himself,
so that, if any of you does evil in ignorance, and thereafter repents
and does righteous good deeds (by obeying Allâh),then surely, He is
Oft­Forgiving, Most Merciful.
Hadith - Sahih Muslim 96, Narrated Abu Hurayrah
The Messenger of Allah, observed: You shall not enter paradise so long
as you do not affirm belief (in all those things which are the
articles of faith) and you will not believe as long as you do not love
one another. ShouldI not direct you to a thingwhich, if you do, will
foster love amongst you: (i.e.) give currency to (the practice of
paying salutation to one anotherby saying) as-salamu alaykum.