Fatoum Al-Jassem, a Syrian girl in the city of Rakka, apparently ran afoul of Islamic law and Al Qaeda in Iraq (known as ISIS or ISIL). Taken to a Sharia court, the judge ruled that having a Facebook account was Zina or immoral behavior and penalized the same way as adultery. (In Islam, many acts of immorality are referred to by a word that translates as adultery.)
The Facebook account was described as an act of great wickedness that merited severe punishment. Hypocritically the Al Nusra Front has a Facebook page and ISIS probably does too. But Muslim men tend to have different standards for women.
Rakka was the first provincial capital to fall to the Sunni Islamic rebels and was captured by the Al Nusra Front, which is the Syrian Al Qaeda affiliate (though these days there is more than just one.)
Al Qaeda in Iraq had developed its own abusive presence in Rakka, vandalizing churches and raising Islamic Caliphate flags over them in triumph. Some of the uglier videos in Syria have come out of the city so this story is not too surprising.
While the only English language report of this appears in Iran's Fars News, it's been widely published in Arabic. ISIS is widely hated now even within the Sunni opposition and so its opponents would have every incentive to make up stories about it. As I have said in the past, so much propaganda is coming out of all the different factions and sides in Syria that it is effectively impossible to determine whether anything is true or not.
The photo used by way of illustration on many of the stories, including this one, appears to be from the film The Stoning of Soraya M about the stoning of a woman in Iran.
Islamic positions on Facebook have been ambiguous in both Sunni and Shiite circles. Some Sunni clerics have issued bans against Facebook for reasons of immorality.