Ubayy bin Ka’ab (RA) stated: 
Allah’s Messenger (PBUH) said: “O Abu al-Mundhir, do you know the verse from the Book of Allah which, according to you, is the greatest?” I said: “Allah and His Apostle (PBUH) know best.” He again said: “Abu al-Mundhir, do you know the verse from the Book of Allah which, according to you, is the greatest?” I said: “Allah, there is no god but He, the Living, the Eternal.” Thereupon he struck me on my chest and said: “May knowledge be pleasant for you, O Abu al-Mundhir!”
Ayat al-Kursi refers to the 255th verse of Surah al-Baqarah. This article provides full translation and tafsir of Ayat al-Kursi.
Here is the full Arabic text of the verse:
Allah! There is no deity except Him, the Alive, the Eternal. Neither slumber nor sleep overtakes Him. Unto Him belongs whatsoever is in the heavens and whatsoever is in the earth. Who is he that intercedes with Him except by His leave? He knows that which is in front of them and that which is behind them, while they encompass nothing of His knowledge except what He will. His throne includes the heavens and the earth, and He is never weary of preserving them. He is the Sublime, the Tremendous.
This verse describes the Oneness of Allah and His attributes in a unique manner. Let us look at Ayat al-Kursi in detail.
Allah! There is no deity except Him, the Alive, the Eternal.
This statement means that there is no god but Allah, and He is the Supreme Lord of all creations/worlds. Also, there is absolutely nothing worth worshipping, except Him. In Surah an-Nisa, Allah says: 
Surely Allah does not forgive that anything should be associated with Him, and forgives what is besides that to whomsoever He pleases; and whoever associates anything with Allah, he devises indeed a great sin.
The Alive, the Eternal means that Allah is Self-Existing (i.e. nothing created him) and that He is Ever Living and Everlasting, Who never dies, Who sustains everyone and everything. All creation stand in need of Allah and totally rely on Him, while He is the Most Rich, Who stands in need of nothing. Nothing can exist without His Order and Will. In Surah ar-Rum, Allah says: 
And of His signs is this: The heavens and the earth stand fast by His command.
Neither slumber nor sleep overtakes Him.
Allah is above and beyond all states of drowsiness or sleep. Allah could never be negligent, unaware, nor can He err in regards to His creation. Rather, He is Supreme and is Aware of what every soul earns. Among His perfect attributes is the fact that He is never affected by slumber or sleep. His power is Absolutely Perfect. In Surah Qaf, Allah says: 
And certainly We created the heavens and the earth and what is between them in six periods and there touched Us no fatigue.
Unto Him belongs whatsoever is in the heavens and whatsoever is in the earth.
This indicates that everyone is a servant of Allah, under His Power and Authority. Everything on earth or in the heavens is owned by Allah. He is the Ultimate Authority. In Surah Ta-Ha, Allah says: 
Unto Him belongs whatsoever is in the heavens and whatsoever is in the earth, and whatsoever is between them, and whatsoever is beneath the soil.
Who is he that intercedes with Him except by His leave?
Nobody can intercede in His Presence except by His permission on the Day of Judgment. The pagans used to think that their idols would intercede on their behalf; this is why Allah has explained that no intercession will work in His Court except which He permits. In Surah al-Anbiya, Allah says: 
And they cannot intercede except for him whom He accepts.
Whereas (except by His leave) refers to Intercession of Prophet Muhammad (PBUH), intercession of some prophets, angels and intercession of some Muslims which they will do for some others.
Narrated by Abu Hurairah (RA): 
Allah’s Apostle (PBUH) said, “For every Prophet there is one invocation which is definitely fulfilled by Allah, and I wish, if Allah will, to keep my that (special) invocation as to be the intercession for my followers on the Day of Resurrection.”
He knows that which is in front of them and that which is behind them,
This refers to His Perfect Knowledge of all creation; including the past, present and future. This is proof of Allah’s Knowledge that encompasses all the worlds. In Surah Ta-Ha, Allah says: 
He knows (all) that is before them and (all) that is behind them, while they cannot compass it in knowledge.
while they encompass nothing of His Knowledge except what He will.
This part of the verse asserts the fact that no one attains any knowledge except that which Allah allows. All-Encompassing Knowledge of every particle in the universe is a unique attribute of none but Allah.
In Surah al-Jinn, Allah says: 
He (alone) is Knower of the unseen, and He does not disclose His (knowledge of the) unseen to anyone, except a messenger whom He has chosen.
His throne includes the heavens and the earth,
Literally, al-Kursi means the “Footstool”, but here the word Kursi represents the Throne of Allah. It is also referred to as Power or Knowledge by some translators of The Quran, like Shakir or Dr Ghali. The reality of His Attributes is above and beyond human reason. This magnificent verse explains Allah’s Existence, Sovereignty, Power and Knowledge that extends over the heavens and the earth.
and He is never weary of preserving them.
Allah has no trouble in preserving and managing whatever is in heavens and the earth, and those who are in the Dominion of Allah. Rather, this is an easy matter for Him.
He is the Sublime, the Tremendous.
Allah is the Most Exalted and the Greatest. All honour, power and superiority belongs to none but Allah. He is the Most High, the Greatest. There is no deity worthy of worship except Him, and no lord other than Him. In Surah al-Imran, Allah says: 
There is no god but He, the Exalted in Power, the Wise.”
Thus, Ayat al-Kursi gives a summarized description of Allah’s Oneness and His Perfections.
- Sahih Muslim Book 04, Number 1768
- The Quran 04:48 (Surah an-Nisa)
- The Quran 30:25 (Surah ar-Rum)
- The Quran 50:38 (Surah Qaf)
- The Quran 20:06 (Surah Ta-Ha)
- The Quran 21:28 (Surah al-Anbiya)
- Sahih Bukhari Volume 9, Book 93, Number 566
- The Quran 20:110 (Surah Ta-Ha)
- The Quran 72:26-27 (Surah al-Jinn)
- The Quran 03:18 (Surah al-Imran)