One of the older, more well-known terrorist organizations is Al-Qaida. You may remember them for their notorious 9/11 attacks in 2001 spawning US military deployment in the region which still exists today. Another household name affiliated with Al-Qaida is Osama Bin Ladin, the mastermind behind the 9/11 attacks. Osama was the center of a world-wide manhunt spanning 10 years only to be caught and killed by US special forces.
Besides Al-Qaida, ISIS, also known as the Islamic State, has taken the spotlight in recent years. With massive global recruiting and previous world-wide terror groups pledging to the ISIS cause, the Islamic State looks like a very formidable foe at the moment.
Finally, some of the older organizations the US recognizes as a terrorist threat are Hamas and Hezbollah. Located in Pakistan and Lebanon these two groups have been fighting western influences for regional control the past few decades.