Terrorism in its modern, recognizable form has existed since the late 1800s. Broadly speaking, there are three phases that terrorism went through, first the "private" phase, second the "state-sponsored" phase and finally the globalized phase.
It was first used by various anarchist revolutionary groups against the state. Anarchists of the late 19th century believed that they were committing what they called "propaganda of the deed" and that just as gunpowder ended the feudal era, the dynamite will end the capitalist one. They believed that the general population was a gunpowder barrel of potential revolutionaries and that their violence would be the spark to ignite this potential: and hence to the downfall of the state. Anarchists carried out bombings of cafes, government buildings, and symbols of financial/business interests. The most infamous example of this in the United States would be the bombing of Wall Street in 1920. Anarchists also carried out assassinations, killing dozens of heads of states including the Russian Czar Alexander II and American president William McKinney. By and large, those were committed by private individuals and organizations.
In the 20th century terrorism was used as a tactic by both left and right wing political groups, however they were most prominently left-wing which became widespread during the Cold war era. State sponsored terrorism became common as the cold war superpowers sponsored groups with ideological affinity to their side. In Europe, you had Soviet support of groups such as the Red Brigades in Italy and the Red Army Faction in Germany. In Latin America, various left-wing revolutionary groups, such as FARC in Colombia, the Shining path in Peru, the ERP in Argentina etc all used terrorism against both military and civilian targets. In Asia, you had Maoist groups such as the Naxalites and Japanese red army. In the US, you had groups such as the weather underground which bombed the Pentagon and the SLA which kidnapped the heiress Patty Hearst. In Canada, you had the FLQ (a left-wing Quebec nationalist group) which kicked off the October Crisis by kidnapping 2 government officials. They caused the only instance when martial law was proclaimed in Canada since 1945. The issue of Palestine also became important and many of those groups committed airline hijackings in the name of Palestinian freedom.
Islamic terrorism came into prominence at the end of the 1970s with the Iranian revolution and the Grand Mosque Seizure in 1979. In the 1980s however it was broadly Shiite militant groups which were at odds with the west.
All those groups used broadly similar tactics: bombings, kidnappings, assassinations, hijackings and sometimes a big seizure of a major site or symbol like the Grand mosque seizure in Saudi Arabia or Palace of Justice siege in Colombia. Those groups which resided in the countryside also committed mass executions at times in order to terrorize the peasantry into supporting them.
After the end of the cold war terrorism became globalized, groups like Al-Qaeda carried out attacks in the US, and it is suspected that Hezbollah carried out bombings against Synagogues and Israeli embassy in Argentina. Funding for those groups also became globalized, with money and personnel flowing through international financial systems and airlines.
Terrorism in many ways is an outgrowth of the economic and political developments in the world. It is an inevitable consequence of easily available weapons which are capable of inflicting disproportionate amount of casualties such as explosives, and of mass literacy which give certain people the motivation to commit politically motivated attacks. Terrorism underwent 3 phases paralleling economic development in the industrialized world, first it was private, then it was state sponsored, and finally globalized. The ideological motivation behind terrorists also changed with the times, in the 60s-70s it was motivated by left-wing beliefs which has since fell out of favor.
However, terrorism, while an ever present threat, should be kept context, terrorists have rarely manage to win a victory against a government throughout their existence. A determined government can almost always defeat a terrorist groups even if it takes decades.