From the root sin-lam-mim (س ل م), which has the following classical Arabic connotations: to be peaceful, quiet, tranquil, content, friendly, reconciled, to be free from imperfections, free of faults, complete, whole, sound, to be safe, secure, well, healthy, and prosperous.

The Salam:
We know from our greeting one another with assalamualaikum, that we are sending peace upon the receiver. As-Salam refers to the attribute of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as the source of this peace. The word Islam also shares the same root of s-l-m which encompasses the concept of peace which is one of core principles of the religion. This name also implies the idea of "soundness" or "wholesomeness," i.e., being flawless, free from blemishes. He is complete and whole. Nothing depletes Him as He is self-sufficient. It is because of this nature of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ the one who is with him can feel safety and security.

In the scripture genesis 2:2, it states, "By the seventh day God had finished the work he had been doing; so on the seventh day he rested from all his work." But this view is inconsistent with the Islamic attributes of God. In Surah Qaf verse 38, "And We did certainly create the heavens and earth and what is between them in six days, and there touched Us no weariness." In this way As-Salam is similar to the meaning of Al-Quddus which both negate any kind of imperfection in His qualities.

Mentions of As-Salam in the Qur'an:

هُوَ ٱللَّهُ ٱلَّذِي لَآ إِلَٰهَ إِلَّا هُوَ ٱلۡمَلِكُ ٱلۡقُدُّوسُ ٱلسَّلَٰمُ ٱلۡمُؤۡمِنُ ٱلۡمُهَيۡمِنُ ٱلۡعَزِيزُ ٱلۡجَبَّارُ ٱلۡمُتَكَبِّرُۚ سُبۡحَٰنَ ٱللَّهِ عَمَّا يُشۡرِكُونَ

Huwal-laahul-lazee laaa Ilaaha illaa Huwal-Malikul Quddoosus-Salaamul Muminul Muhaiminul-aAzeezul Jabbaarul-Mutakabbir; Subhaanal laahi Ammaa yushrikoon

English Translation:
"Allah is He, than Whom there is no other god;- the Sovereign, the Holy One, the Source of Peace (and Perfection), the Guardian of Faith, the Preserver of Safety, the Exalted in Might, the Irresistible, the Supreme: Glory to Allah! (High is He) above the partners they attribute to Him."— (Qur'an 59:23)

The usage of the word Salam is used repeatedly throughout the Qur'an but referring to the attribute of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as the giver of peace only once. For example, in Surah as-Saffat verse 181:

وَسَلَٰمٌ عَلَى ٱلۡمُرۡسَلِينَ

Wa salaamun ‘alalmursaleen

English Translation:
"And peace upon the messengers."— (Qur'an 37:181)

But remember the originator of such peace is Allah. By turning to Him by this name As-Salam, seeking His guidance, we hope to attain the final destination in paradise (home of salam, peace).

From Surah Yaseen:

سَلَٰمٞ قَوۡلٗا مِّن رَّبّٖ رَّحِيمٖ

Salaamun qawlam mir Rabbir Raheem

English Translation:
"[And] “Peace,” a word from a Merciful Lord."— (Qur'an 36:58)

We provide a little more context here with the previous three ayat leading up to verse 58, "55. Verily, the dwellers of the Paradise, that Day, will be busy with joyful things. 56. They and their wives will be in pleasant shade, reclining on thrones. 57. They will have therein fruits and all that they ask for. 58. (It will be said to them): “Salam (Peace!)” — a Word from the Lord, Most Merciful."

In the hadith, the Prophet Muhammad (ﷺ) affirmed this name when he heard the companions (may Allah be pleased with them) say, "salam upon Allah." He corrected them saying, “Allah Himself is As-Salam (Peace), and if anyone of you prays then he should say, 'All the compliments, prayers and good things are due to Allah; peace be on you, O Prophet, and Allah's mercy and blessings [be on you]. Peace be on us an on the pious subjects of Allah.'”

Supplicate with this name:
In another narration, it was reported that the Prophet (ﷺ) recited this name in his supplication after prayer. Thauban reported: When the Messenger of Allah (ﷺ) finished his prayer. He begged forgiveness three times and said: (Allahumma Antas-Salam wa minkas-salam. Tabarakta ya Dhal-jalali wal-Ikram.) O Allah! Thou art Peace, and peace comes from Thee; Blessed art Thou, O Possessor of Glory and Honour. Walid reported: I said to Auza'i: How is the seeking of forgiveness? He replied: You should say: I beg forgiveness from Allah, I beg forgiveness from Allah."

Reflection:
We can benefit from knowing As-Salam by working daily to attain peace. This process involves freeing the heart and mind from silly games with no upside, like jealousy, envy, and revenge. As the saying goes, "resentment is like drinking poison and then hoping it will kill your enemies." This is not the way to achieve peace. Be forgiving of yourself and others. Focus your effort to try and become as close to flawless as you can get. Work to minimize all your imperfections. When a person can love himself and love others, he will naturally bring peace to the earth.

The Prophet ﷺ also characterized the believer as a being safe. It was narrated from Abu Hurairah that: The Messenger of Allah ﷺ said: “The Muslim is the one from whose tongue and hand the people are safe, and the believer is the one from whom the people's lives and wealth are safe.”

Anas reported that the Messenger of Allah, may Allah bless him and grant him peace, said, “As-Salam (peace) is one of the Names of Allah Almighty which Allah has placed in the earth. Therefore give the greeting among yourselves” and “Abu Huraira reported: The Messenger of Allah (may peace and blessing be upon him) 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. Should I not direct you to a thing which, if you do, will foster love amongst you: (i.e.) give currency to (the practice of paying salutation to one another by saying) as-salamu alaikum.”

References:
01) Sahih al-Bukhari 831
02) Sahih Muslim 591
03) Sahih (Darussalam) Sunan an-Nasa'i 4995
04) Hasan (Al-Albani) Al-Adab Al-Mufrad 989
05) Sahih Muslim 54a