The group started out as Ama'at al-Tawhid wal-Jihad, but this was changed after announcing support of Osama Bin Lade to the more catchy and shorter AQI (al-Qaeda in Iraq)
The group became ISI (Islamic State in Iraq) in 2006 and then added "al Sham" to become the ISIS in 2013 (Note this is important later.)
ISIS (The Islamic State of Iraq and Syria) dropped the last two letters to become IS (Islamic State) in the June 2014 branding of the organisation.There is some dispute as to whether this is an fact "Syria" as we know it or the wider ancient Syrian empire
The US branding prefers to be ISIL (Islamic state of Iraq and Levant) as the the US believes the historic phrase "Levant" more closely matches the new declared "State" now includes Syria, Lebanon, Israel, Palestine and Jordan, and a closer translation of "al Sham (as introduced in 2013, and also present in the Arabic translations)
In the meanwhile the French refer to them as "EIIL" ( L'Etat islamique en Irak et en Syrie)) or rather did until they decided that in line with many Arabic news broadcasters they would now refer to the organisation as DAIISH or Da'esh (abbreviation of Dawlat al-Islamiyah f'al-Iraq wa al-Sham)