The Euskadi ta Askatasuna or ETA, were founded in the Spanish Basque country in 1959, during the repressive Franco regime, as a radical offshoot of the more moderate Partido Nacionalista Vasco, or Basque Nationalist Party. The ETA's name is variously translated as "Basque Homeland and Freedom" and "Basque Fatherland and Liberty". Their goal was an independent and sovereign Basque homeland (Euskadi, in Basque), with its capital at Guernica, and their method was violence.
The ETA started its campaign with a series of bombings in cities throughout the Basque Country, and a failed attempt to derail a train carrying fascist veterans of the Spanish Civil War who were heading to a reunion. The Franco Government's response, predictably enough, was to crack down violently, and most of the ETA fled the resulting torture and reprisals into the exile of the French Basque country. From there they carried on a low-level campaign of occasional bombings and assasinations, probably with a certain level of tacit support from the local population, if only because of the repressive nature of the fascist Franco government.
The turning point came in 1976. Franco died, and a new, democratic government was established in Spain. The new government met most of what had been the ETA's demands. Spain underwent considerable devolution during the period, and the Basque country was given its own, relatively powerful local parliament, and the Basque language was no longer suppressed. At this point, the ETA returned to Spain and established its political wing, the Herri Batasuna (Sinn Fein to the ETA's IRA), but most of their impetus for existence had fled.
During the late '70s and '80s, the ETA dwindled to a radical core of die-hards who were willing to settle for nothing less than a totally independent and sovereign Basque nation, and were prepared to act more and more violently to achieve this. They gradually alienated much of their popular support, and in 1983 a new group, the GAL, or Anti-Terrorist Liberation Group, appeared, that had as its sole raison d'etre the hunting down and killing of ETA members. The GAL was almost certainly a creation of the new Spanish government, though this was never conclusively proved.
In the late '80s and through the '90s, the ETA declared and broke a series of truces and cease-fires, and had its popular support dwindle to almost nothing. The Herri Batasuna had terrible results in the 1996 Basque country elections, getting less than 20% of the local vote, and in 1997 it was dissolved entirely, when, in the aftermath of the particularly unpopular ETA assassination of a local Basque leader, proof surfaced which seemed to conclusively prove the Herri Batasuna was an arm of the ETA.
The ETA still exists today, and is still nominally active, though it probably only consists of a few dozen active members and a few hundred more supporters.