A couple of things I want to mention as well that I believe /u/JustBecauseOfThat did not mention.
ISIS/Al-Qaeda were at one point the same organization. ISIS used to be Al-Qaeda in Iraq until they became ISIS. So they hold the same ideology...more or less. We'll get into how this ideology works in the real world and where ISIS became this terrible organization that seems to have surpassed Al-Qaeda in terms of brutality. More on the ideology in a bit.
Al-Qaeda lives in the real world...So they still kill people, exploded bombs in public, use suicide bombers etc. etc. BUT they realize that they cannot kill indiscriminately and have to be selective. They also want people to follow them, not just those who join up but also the moderates and eventually all Muslims...Minus the Shia of course...So they can't risk pissing off people too much.
Here is the gist of Al-Qaeda's grand twenty year plan.
First phase 2000-2003: Provoke America into invading...America invades and Muslim world under attack...you have Muslim awakening. (It didn't go down like that). Also in 2001, a guy named Abu Musab al-Zarqawi moves from Afghanistan to Iraq and sets up Al-Qaeda in Iraq which will later become ISIS.
Second phase 2003-2006: Al-Qaeda becomes a wider movement spreads to Iraq, and specific Gulf areas. The important thing to understand about Zarqawi is that he didn't want to attack the coalition forces in Iraq, rather he wanted to attack the Shia and anyone else that didn't conform to his ideology (more on that in a bit). This guy is also Baghdadi's (current ISIS leader) mentor.
Third phase 2007-2010: More attention on Syria and Iraq
Fourth phase 2010-2013: Arab governments collapse because of their efforts. Now we all know that these terrorists were not the reason the Arab spring occurred. But in their delusions they believed it was all them. Its around this time that Baghdadi took control.
Fifth phase 2013-2016: Al-Qaeda would declare the caliphate. BUT OSAMA BIN LADEN GOT KILLED. Ayman al-Zawahiri took control and he is not charismatic, skilled at strategy...so there were disagreements between Baghdadi and Zawahiri. So ISIS instead declares a caliphate after taking over many cities in Syria and Iraq.
There are two more phases but not important.
IDEOLOGY
Both ISIS and Al-Qaeda are from a sect of Sunni Islam called Salafism. So what is Salafirm?
"Salafism derives from the term the pious forefathers (al-salaf al-salih), the first three generations of Muslims who had first-hand experience of the rise of Islam and are regarded as exemplary for the correct way to live for future Muslims." Roel Meijer
The period which they are talking about lasts from the year 610 to the year 855. So this is where the fighting comes in. Islam at the time was a young religion and Jihad was necessary to make it bigger and defend against those that wanted to end it.
It doesn't really apply now because there are 1.6 Billion Muslims, but to these terrorists, their understanding of Islam which is over 1000 years old is under threat so they are going to fight to keep it "pure".
Now Salafism itself is harmless, it would be like Ultra orthodox Christianity or Judaism. There are Salafist who peacefully live their lives and follow old traditions without getting into conflict.
The terrorist ideologues (Abu Muhammad al-Maqdisi and a few others) came up with Jihadi-Salafism which was all about fighting to keep everything as original as possible and that's how you have the birth of Al-Qaeda and eventually ISIS.
Now why did it become so popular? Well let's look a the Middle East, tons of oppressive governments and they maintain control through torture, execution and just strict control. Those living abroad coming to fight are youths that don't fit into the society.
So if you are a poor person that has had his life ruined by the government, suddenly you join this group and you get a gun, brothers to fight along with, food, security. YOU ARE NO LONGER A NOBODY, YOU ARE POWERFUL.
And that is why ISIS has picked up so much momentum. In Syria, you have Assad, no explanation needed as to why people don't like him.
In Iraq, you had the Maliki government, in front of the United States they were all about playing fair but as soon as the U.S. they continued on doing the same shit Saddam Hussein was doing. So suddenly you are this person, constantly getting bullied by the government, discriminated against, can't get a job, can't get food, can't get an education. So ISIS becomes the only option available. Again...YOU ARE NO LONGER A NOBODY, YOU ARE POWERFUL.
Now the thing that is messed up about Jihadi-Salafism is that it ignores everything that has happened since the year 855...so collectively as Muslims, many things were decided were no longer applicable. Because society has progressed...social, political, economic, systems that applied back then, no longer apply now. But ISIS says NOPE. We are going to stick to the original as possible, and we are going to kill anyone that doesn't agree.
Now coming back to Al-Qaeda...they live in the real world, they know they can't piss off all the Muslims.
ISIS doesn't give a shit...and hence their campaign of brutality and they gained supporters, because it gives them power. They are no longer nobody's at the mercy of the oppressive government.
Now Maqdisi the guy who originally came up with Jihadi-Salafism...disagrees with what ISIS is doing. So he is now considered a disgrace by ISIS.
Instead they created their own ideologue's who are now justifying all the rape and murder.
Now /u/JustBecauseOfThat provides the rest in a good summary of what Islam allows and forbids.