Declension wrote:
A perceived problem, and a belief that non-terrorist options cannot solve the problem.
This.
Under the common, implicit definition of terrorism, i.e. acts of violence organised by a group (*) of people with the purpose of coercing another group or a government into changing a present policy or adopting a new one, it happens when a group feels strongly enough about a cause (which happens often), and when there are no reasonnable ways of pursuing the change through legal means -- even if legal change is unlikely, its virtual possibility is important.
In the specific context of whatever acts of terrorism are committed by Middle Easterners against the USA, the cause is repeated American intervention in the area. Obviously, they can't petition to the Congress or President, they can't vote in the next election, and their governments cannot openly confront the United States, so... there you go, terrorism.
(*) I think "organised" and "group" are key terms, if we want to keep the concept of terrorism as clearly defined as possible. This way, the Norwegian bomber/mass murderer isn't a terrorist, because even though he had planned his action, he was not a group -- and positively changing a policy can't be done alone, since a single person will be arrested very quickly, which is not good for the endurance of the cause. He is just a murderer, probably a crazy one too. In the same way, a mob which commits vandalism during a riot, even a riot degenerating from a legal protest organised by a group, would not be committing terrorism, since the actual violent acts were not organised. If the distinction isn't made, "terrorism" becomes confused with "politically-flavoured violence".