Terrorism has existed as long as there have been rebellions. You could look at any form of Guerrilla tactics as a form of terrorism. The Boston Tea Party could be seen as an act of economic terrorism. Viking raids can be seen as an act of terrorism. So while the concept of terrorism has existed for as long as people have feared each other (with bands raiding other bands as probably the earliest true form of terrorism); Modern Terrorism (which is what i believe you are alluding to) would probably be linked to the Balkan Region in which terrorist groups at the time were simply independence factions trying to make life more difficult for their oppressor. Be it an Armenian revolutionary trying to pass inflamatory pamphlets, smuggle arms for themselves and others like them, or whatever means of sabotage. A more individualized version could be John Brown's Raid on Harper's Ferry. Gavrilo Princip at Sarajevo is another big example. Thus modern terrorism probably took form in the late 19th-early 20th century.

A common theme amongst the early "terrorist" (and I use quotations because we would now call them revolutionaries which use similar methods, but different in their long term goals) in Ireland and Palestine Post World War I with both parties seeking indpendence. This idea would be the foundation of most terrorist groups going through WWI/II occupation & beyond in colonial regions seeking independence. Communist terrorist groups/Western backed terrorist groups would also be thrown in the mix as well with the communist groups ideas being fairly obvious (like the Cubans) and with western backed ones having their own reasons and the thing that they shared was that they were backed by a western power and had a reason to hate communism (think the Mujahadeen in Afghanistan during the invasion; very different from what most american's have in terms of ideology yet they both wanted a non-communist afghanistn (hell they both didn't even agree on whether or not it would be "free")). After communism fell and it was clear that was a failed economic system, the face of terrorism shifted into what we all would recognize as religious fanatics more or less using the same tactics (with alterations given new technology) as the Armenian's almost 100 years ago. So while the people wielding the tactics have changed dramatically, the tactics themselves haven't seen as radical a shift as, say, conventional warfare had in that same period.

Edit: this is not to say that all terrorist groups are religious fanatics; some are still independence factions seeking independence for a minority (the Kurdistan Workers' Party is a good example, also Hamas/Hezbollah are a poorer example since they are also partially motivated by a fanatical hatred of Israel)