http://www.vox.com/2015/11/14/9734894/daesh-isis-isil
There are, broadly speaking, four things that people call the group: ISIS, ISIL, the Islamic State, or Daesh. This is largely ISIS’s fault; a big reason the group has so many names is that it keeps changing it.
When the group’s predecessor organization was created in 1999, it was called Jamaat al-Tawhid wal-Jihad, which means Unity and Jihad. In 2004, the group’s founder, Abu Musab al-Zarqawi, pledged an oath to al-Qaeda, changing his group’s name to Tanzim Qaidat al-Jihad fi Bilad al-Rafidayn — or, as it was called in English, al-Qaeda in Iraq.
After AQI took over huge swaths of Iraq in 2006, the organization declared itself to be a state in northern Iraq, and started calling itself the Islamic State in Iraq. When it took a bunch of territory in Syria in 2013, it began calling itself the Islamic State in Iraq and al-Sham — ISIS.
Al-Sham is a difficult-to-translate Arabic term referring to a specific geographic area along the eastern Mediterranean that includes Syria. Some English speakers translate al-Sham as “the Levant,” which refers to a broader region in the Middle East that generally overlaps with al-Sham. This is how you get ISIL (Islamic State in Iraq and the Levant), as the White House and others call it. Others still approximate al-Sham to Syria, which yields the same ISIS acronym.
The full name in Arabic is transliterated like this: al-Dawla al-Islamiya fi Iraq wa al-Sham — which produces the acronym DAIISH (usually spelled Daesh in English). That sounds an awful lot like the Arabic word “dahes,” which the Guardian translates as “one who sows discord.” ISIS kind of hates this insulting connotation, and so banned the name “Daesh” in its territory.
But it doesn’t use ISIS either. Crucially, the group now claims to be a caliphate — that is, the successor of the original seventh-century founding Islamic nation. As such, it dropped the geographic identifiers from its name, and simply calls itself “the Islamic State.”