What is the Black Stone?
The Black Stone (called الحجر الأسود al-Hajaru-l-Aswad in Arabic) is a Muslim relic, which according to Islamic tradition dates back to the time of Adam and Eve. Many geologists and historians believe it to be a tektite or a meteorite.It is the eastern cornerstone of the Kaaba, the ancient sacred stone building towards which Muslims pray, in the center of the Grand Mosque in Mecca, Saudi Arabia 
Do Muslims Lick a Black Stone?
A Christian Missionary is trolling internet forums and comment sections and declaring Muslims to be “black stone lickers”
This is obviously a lie, a malicious one at that. There is no injunction from the Quran, Hadith literature or work of Fiqh which commands Muslims to lick any black stone. Quite simply this Christian missionary is using his fertile imagination and making things up.
Muslims do Kiss (not lick) al-Hajaru-l-Aswad (Black Stone) During Hajj
The black stone (al-Hajaru-l-Aswad) is located at the outer corner of the Kaba (cube-shaped building in Mecca) and serves the function of being an indicator of where circumambulation (Tawaf) of the Kaba commences and ends. Muslims, during the pilgrimage to Mecca (Hajj), perform Tawaf around the Kaba seven times as a rite of Hajj.
This stone is kissed prior to commencing Tawaf (circumambulation) and each time one rounds the Kaba it is recommended the black stone is kissed (if possible). 
Why do Muslims Kiss the Black Stone?
It was a practice of Muhammad (pbuh) and Muslims do it in order to emulate the last Prophet of God (Muhammad, pbuh).
Do Muslims Worship the Black Stone (al-Hajaru-l-Aswad)?
Muslims do not worship this stone. Kissing does not constitute worship. In fact Umar, the second Khalifah and companion of the Prophet Muhammed (p), made this perfectly clear when he said:
Umar came near the Black Stone and kissed it and said "NO DOUBT, I KNOW THAT YOU ARE A STONE AND CAN NEITHER BENEFIT ANYONE NOR HARM ANYONE. Had I not seen Allah's Apostle kissing you I would not have kissed you." 
This clearly shows us the back stone is not meant to be worshipped by Muslims. In fact this would negate a person’s iman (belief/faith), i.e. if a Muslim did worship the stone then he/she would be considered non-Muslim as only Allah has the right to be worshipped.
A Malicious Attack on the Black Stone: “It is looks like a Vagina”
Some Islamophobes have begun a baseless attack on the appearance of the black stone in order to insult Muslims.
How a stone looks like a vagina to them is beyond me. There is a silver frame around the black stone. This silver frame is a modern addition and does not constitute the black stone but with the addition of the silver frame the structure does look doughnut shaped. The claim of vagina-resemblance is simply a product of their debauched and malicious minds.
As Muslims (God’s chosen people) we are accustomed to people looking to insult us; a sad sign of the times.
Was the Black Stone Worshipped by Arab Pagans Prior to Islam?
I am not sure if it was, the Arab Pagans did have 360 idols which they housed at the Kaba and the Prophet destroyed ALL their idols. 
However, even if some of them did worship it we must recognise Islam came to abolish such practices. Therefore it has no bearing on Islam.
The black stone was originally given to Abraham who built the Kaba used the black stone as a corner stone for the Kaba. Abraham and Ishmael taught the early Meccan Arabs monotheism; after the passing on of Abraham and Ishmael the Meccan Arabs, with time, regressed into pagan practices such as idol worship and they ended up having a pantheon of gods despite the original message of Abraham and Ishmael which taught the early Arabs to worship Allah alone (monotheism) .
Prophet Muhammed (p) was sent to bring these Arabs back to the teachings of Abraham and outlaw polytheistic and pagan practices (i.e. purify the teachings of Abraham from the polytheistic practices which were innovated into the teachings with time) . Thus any history of the black stone being worshipped by pagan Arabs has no bearing on Islam whatsoever.
 Reliance of the Traveller, Ahmad ibn Naqib al Misri, edited and translated by Nuh Ha Mim Keller, Amana Publications, 1994 (J5.2,J5.14)
 Sahih BUKHARI: Volume 2, Book 26, Number 667
 Ar-Raheequl-Makhtum by Safi-ur-Rahman Al-Mubarakpuri, Darussalam, 2002 pg 26-28
Here is a good video from Iqraproductions educating people about al hajr ul aswad (black stone) clearing up some misconceptions related to the black stone as well as explaining the history behind it:
What is the Black Stone?
Do Muslims Worship the Black Stone?
Was it considered to be the genitalia of a pagan goddess by the pagans of Mecca?
Why do Muslims kiss the Black Stone?
What is the Kaba? Muslims do not worship the Kaba (explained by Dr Zakir Naik):