When Islam came, Allaah prescribed that the manner of greeting among Muslims should be “assalaamu alaykum (السلام عليكم)". This is the general form of greeting used in Muslim lands and nations, even though local influence may have caused it to modify it or even distort it, for instance saying 'slamalaikum' which do not have any useful meaning as far as I know.
Meaning of assalamu alaikum
Salaam (سلام) is literally translated into 'peace'. When the definite article al- is used, as in al-salaam which is pronounced as 'as-Salaam', then that is one of the beautiful names of our Lord, Glorified be He, free from all imperfections and need. This is one of the glorification that Muslims can say verbally after completing the obligatory prayer (Salaah/Namaaz), as reported in Saheeh Muslim (1/414) from the words of the Prophet of Allaah, peace and blessings be on him, "اللهم أنت السلام ، ومنك السلام" meaning "O Allaah, You are the Peace and from You is all peace (meaning, all peace and tranquility only comes through You)".
The usage of the preposition ‘ala in ‘alaykum (upon you) indicates that the greeting is inclusive. The plural pronoun (-kum) implies everyone if it is in a group or family. Hence the phrase assalamu alaykum translates into 'peace be on you all' and it is an invocation to the Lord for safety and harmlessness for the person (to which assalamu alaykum is said).
Importance for saying assalamu alaikum
The companion 'Abdullaah ibn ‘Umar , may Allaah be pleased with both 'Abdullaah and 'Umar, reported that a man asked the Messenger of Allaah (peace and blessings of Allaah be upon him):
“What is the best thing in Islam?” He said, “Feeding others and giving the greeting of salaam to those whom you know and those whom you do not know.” (Saheehayn and others)
((أن رجلا سأل رسول الله صلى الله عليه وسلم : أي الإسلام خير ؟ قال : تطعم الطعام ، وتقرأ السلام على من عرفت ومن لم تعرف))
The great Scholar of the past, al-Qaadee 'Ayaad (القاضي عياض) said in Ikmaal al-Mu’allim (1/276):
"Here the Prophet (peace and blessings of Allaah be upon him) was urging the believers to soften their hearts. The best Islamic attitude is to love one another and greet one another, and this is achieved by words and deeds. The Prophet (peace and blessings of Allaah be upon him) urged the Muslims to foster love between one another by exchanging gifts and food, and by spreading salaam, and he forbade the opposite, namely forsaking one another, turning away from one another, spying on one another, seeking out information about one another, stirring up trouble and being two faced.Thus the Prophet (peace and blessings of Allaah be upon him) explained that this salaam spreads love and brotherhood. The companion Aboo Hurayrah (may Allaah be pleased with him) reported that the Messenger of Allaah (peace and blessings of Allaah be upon him) said:
Love is one of the duties of Islam and one of the pillars of the Islamic system. One should give salaams to those whom one knows and those whom one does not know, out of sincerity towards Allaah; one should not try to impress other people by giving salaams only to those whom one knows and no-one else. This also entails an attitude of humility and spreading the symbols of this ummah through the word of salaam.”
“You will not enter Paradise until you believe, and you will not believe until you love one another. Shall I not tell you about something which, if you do it, you will love one another? Spread salaam amongst yourselves.” (Saheeh Muslim and others)
((لا تدخلون الجنة حتى تؤمنوا ولا تؤمنوا حتى تحابوا أولا أدلكم على شيء إذا فعلتموه تحاببتم أفشو السلام بينكم))
al-Qaadee ‘Ayaad said in Ikmaal al-Mu’allim (1/304):
And there are other hadiths that stress the importance of salam in Islam. Such as:
"This is urging us to spread salaam, as mentioned above, among those whom we know and those whom we do not know. Salaam is the first level of righteousness and the first quality of brotherhood, and it is the key to creating love. By spreading salaam the Muslims’ love for one another grows stronger and they demonstrate their distinctive symbols and spread a feeling of security amongst themselves. This is the meaning of Islam.”
On the authority of al-Bara' bin 'Azib (may Allah be pleased with them) the Messenger of Allaah commanded us to do seven things: to visit the sick, to follow the funeral (of a dead believer), to invoke the Mercy of Allah upon one who sneezes, to support the weak, to help the oppressed, to promote the greeting of 'As-Salamu 'Alaikum', and to help those who swear to do something to keep their oaths. (Saheehayn, as quoted in Riyadus Saliheen of Imam an-Nawawi)
In fact, one can see how the great Scholars of Islam stressed the importance of salaam, due to the fact that the Messenger of Allaah stressed it so much. Imam Yahya an-Nawawi, in his widely known book 'Riyadus Saliheen', has dedicated one whole section on greetings, calling it 'The Book on Greetings'. Within this section, he has 13 chapters conveying A-Z about 'salaam' in Islam! This petty post cannot stress enough the importance and blessings of this simple saying: 'Assalamu Alaikum'.
What to say to spread salam?
One can say either 'Assalamu Alaikum (السلام عليكم)' or 'Salamun Alaikum (سلامٌ عليكم)' and they are both correct and acceptable as reported in the authentic texts. What is even better is to say 'Assalamu alaikum warahmatullah/Salamun alaikum warahmatullah(سلامٌ عليكم ورحمة الله)' and the best thing is to say 'Assalamu alaikum warahmatullahi wabarakatuh/Salamun alaikum warahmatullahi wabarakatuh (سلامٌ عليكم ورحمة الله وبركاته)'.
The Prophet (peace and blessings of Allaah be upon him) explained the reward earned by the one who says salaam, as was reported in Saheeh Ibn Hibbaan and others. The companion Aboo Hurayrah (may Allaah be pleased with him) said that a man passed by the Messenger of Allaah (peace and blessings of Allaah be upon him) whilst he was sitting with some others, and said “Salaam ‘alaykum (peace be upon you).” The Prophet (peace and blessings of Allaah be upon him) said, “[He will have] ten hasanaat (rewards).” Another man passed by and said “Salaam ‘alaykum wa rahmatullaah (peace be upon you and the mercy of Allaah).” The Prophet (peace and blessings of Allaah be upon him) said, “[He will have] twenty hasanaat (rewards).” Another man passed by and said “Salaam ‘alaykum wa rahmatullaahi wa barakaatuh (peace be upon you and the mercy of Allaah and His blessings).” The Prophet (peace and blessings of Allaah be upon him) said, “[He will have] thirty hasanaat (rewards).”
Our Lord tells us that the greeting of the angels to the people of Paradise will be “Salaamun ‘alaykum” as He says (interpretation of the meaning):
And angels shall enter unto them from every gate (saying):[Surah ar-Ra’d 13/23, 24]
‘Salaamun ‘Alaykum (peace be upon you) for you persevered in patience! Excellent indeed is the final home!'
((وَالْمَلائِكَةُ يَدْخُلُونَ عَلَيْهِمْ مِنْ كُلِّ بَابٍ سَلامٌ عَلَيْكُمْ بِمَا صَبَرْتُمْ فَنِعْمَ عُقْبَى الدَّارِ))
Saying 'wassalam' at the end of a letter or post
There is nothing wrong with ending a letter with the phrase “wassalaam (والسلام)” and it is not a condition that the phrase should be used in full, because when a written abbreviation is used, what the writer means is the complete expression. So when a person says “wassalam”, what he or she means is “was salaamu ‘alaykum.” But if the person sending the letter writes “wassalaamu 'alayka” or “wassalaamu ‘alaykum” at the end, this is better. The companion 'Umar ibn al-Khattaab, may Allaah be pleased with him, ended his letter to the Qaadee (Judge) Shurayh with the words “wassalaamu ‘alayka” [Sunan an-Nasa'i].
However, the person sending the letter should greet the addressee with the complete phrase (“Assalaamu 'alaykum”) at the beginning of the letter, as has been the habit of the Muslims from the time of the Prophet (peace and blessings of Allaah be upon him) until the present. And Allaah knows best.
Don't be a miser or try to be too correct grammatically!
This is an important reminder. Some Muslims, who are aware that plural pronoun ends with -kum and masculine pronoun ends with -ka and feminine pronoun ends with -ki in Arabic words to refer to a person or people, resort to saying "assalaamu 'alayka or assalamu 'alayki" instead of "assalamu alaykum". While this is not wrong or unacceptable, this is an unnecessary correction that restricts the supplication to only the person that the greeting of salaam is being sent to, and excludes the angels that are present (but invisible) and the person's family.
Yes, you have completed the bare minimum and yes, you have been 'grammatically correct' in your greeting, but please do not exclude my family from your supplication for granting peace and blessings and mercy from the Lord of the Universe! Don't be a miser in your supplication.
Returning the reply to salaam
It is obligatory that you return the greeting of salaam with something equivalent or better. In some cultures, when a lower-class person gives salaam to an upper-class person, the latter do not bother to return the reply. This is contradictory to the teaching of Islaam and we seek refuge in Allaah from such superiority complex.
The Prophet (peace and blessings of Allaah be upon him) commanded us to return salaams, and made it a right and a duty. It is reported on the authority of the companion Aboo Hurayrah (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: “The Muslim has five rights over his fellow-Muslim: he should return his salaams, visit him when he is sick, attend his funeral, accept his invitation, and pray for mercy for him [say “Yarhamuk Allaah”] when he sneezes.” (Saheehayn and others)
Allaah says (interpretation of the meaning):
“When you are greeted with a greeting, greet in return with what is better than it, or (at least) return it equally”
[Surah an-Nisa’ 4:86]
Why say Assalamu Alaikum to the Muslim
Would you not agree that you'd rather have someone supplicate to our Lord for you and your family instead of saying 'hello'? I agree it is probably easier saying 'hi' as it's shorter containing just two alphabets, but if you really wish good for a person, then invoke Allaah's safety and security for the person.
This is something that may not be known or understood correctly by non-Muslims and even many laymen Muslims. Yes, assalamu alaikum is a Muslim thing, but ponder for a moment the beauty of the teaching of Islam that asks one another to supplicate for safety and blessings and mercy and thereby increase love and respect between one another, instead of giving a 'hello' or 'good morning' out of courtesy.
May our Lord increase love and respect between one another and guide us all to that which is true and beautiful.
References: 1, 2 and 3.