Salamun Alaikum (Peace be upon you)
The Arabic word for stoning is 'Rajm'. This subject is dealt with in a question and answer format.
What is Ayat ul Rajm?
A verse regarding stoning to death as a punishment for adultery.
Does Ayut ul Rajm exist in the Quran?
Does the Quran distinguish between fornication and adultery?
What does the Quran refer to then?
Unlawful Sex (Regardless of being married or not) = 'Zina'.
Does the Quran specify punishment for 'Zina'?
What is the Quranic punishment for Zina?
100 lashes after complete proof has been established including those of 4 witnesses to the act (24:2; 24:4; 24:13).
What is the Quranic punishment for one who falsely accuses another of ‘Zina’?
80 lashes and to never accept their evidence again (24:4) unless they repent and mend their ways (24:5).
Does the Quran specify punishment for 'Zina' as stoning?
Does the Quran specify a punishment of stoning for anything?
In what capacity then does the Quran speak about stoning?
When referring to punishments carried out by Pagans (11.91; 18:20; 19.46; 26:116; 36:18) on believers (and in specific cases, threats to Prophets).
Where do we find the directive of stoning to death as a punishment for adultery then?
How is the lack of the stoning commandment in the Quran justified?
This was a verse that once existed but was subsequently lost during the compilation of the Quran.
How was it lost?
A goat possibly ate it.
What informs us that potential verses were lost during compilation or that the stoning verse was eaten by a goat?
The same Islamic secondary sources
The following is given by way of an example of such a narration within Islamic secondary sources. No in-depth analysis of this report within the sixth Sunni canonical hadith literature is intended other than to provide a reference. Various Muslim theologians offer various explanations for and against the authenticity of this narration and its wider corpus (Sunan Ibn Majah). It is left to the reader to accept or reject their explanations based on the merits of their arguments and whether or not the explanation is convincing.
"Reported ‘Aisha: ‘When the verse of stoning and of suckling an adult ten times (Rezah Kabir) was revealed, and they were (written) on a paper and placed under my pillow, when the Messenger of God (pbuh) expired and we were occupied by his death, a domestic goat entered and ate away the paper."
The following hadith from Sahih Bukhari implies that the verse once existed and was recited, but eventually was lost.
Narrated Ibn 'Abbas:
"... Ibn Abbas added: We reached Medina by the end of the month of Dhul-Hijja, and when it was Friday, we went quickly (to the mosque) as soon as the sun had declined, and I saw Sa'id bin Zaid bin 'Amr bin Nufail sitting at the corner of the pulpit, and I too sat close to him so that my knee was touching his knee, and after a short while 'Umar bin Al-Khattab came out, and when I saw him coming towards us, I said to Said bin Zaid bin 'Amr bin Nufail "Today 'Umar will say such a thing as he has never said since he was chosen as Caliph." Said denied my statement with astonishment and said, "What thing do you expect 'Umar to say the like of which he has never said before?"
In the meantime, 'Umar sat on the pulpit and when the callmakers for the prayer had finished their call, 'Umar stood up, and having glorified and praised Allah as He deserved, he said, "Now then, I am going to tell you something which (Allah) has written for me to say. I do not know; perhaps it portends my death, so whoever understands and remembers it, must narrate it to the others wherever his mount takes him, but if somebody is afraid that he does not understand it, then it is unlawful for him to tell lies about me. Allah sent Muhammad with the Truth and revealed the Holy Book to him,and among what Allah revealed, was the Verse of the Rajam (the stoning of married person (male & female) who commits illegal sexual intercourse, and we did recite this Verse and understood and memorized it. Allah's Apostle did carry out the punishment of stoning and so did we after him.
I am afraid that after a long time has passed, somebody will say, 'By Allah, we do not find the Verse of the Rajam in Allah's Book,' and thus they will go astray by leaving an obligation which Allah has revealed..."
Does the Quran indicate that there are any missing verses?
NO - It claims to be a book fully detailed (16.89; 6:114), complete and all that is required for right guidance and religion (7:52).
WHAT IS HALF OF DEATH?
Quite apart from the incredulous claims made by Islamic secondary sources that there were verses that went missing from the Quran, the fact that a stoning verse was never a part of the Quran can be attested by the following verse:
We note from the above verse that the punishment of a wedded woman from the category of 'right hand possess' is half that of a free believing woman who commits the same indecency. If the punishment for adultery for a free woman was ever stoning to death, what would amount to half of death in the case of a married woman from the category of 'right hands possessed' and who was also guilty of adultery?
It is clear that the stoning to death verse was never a directive of the Quran nor did it ever exist as part of it.
(1)Sex with Slave Girls
 SUNAN IBN MAJAH, Book of Nikaah, Number 1944, Hadith in Arabic [online], http://hadith.al-islam.com/Page.aspx?pageid=192&TOCID=638&BookID=29&PID=1934 [Accessed 13th April 2011]
 SAHIH BUKHARI, Volume 8, Book 82, Number 817, Center for Muslim-Jewish Engagement [online], Translation available at http://www.cmje.org/religious-texts/hadith/bukhari/082-sbt.php
Any emphasis in bold black, are my own insertions. They have no bearing on the original text other than they emphasise relevance to the topic at hand. These are merely illustrations and have solely been utilised for educational and explanatory purposes.