The Batavi were a Germanic1 tribe from the Roman province of Belgica, who inhabited an area in the the Rhine delta, in particular the low-lying ground between the rivers Waal and the Rhine, which was therefore known as the insula Batavorum2 or Batavian Island, which of course is now all in the modern Netherlands.
The Batavi were brought under Roman rule during the time of Augustus Caesar in the year 13 AD, and were classed as allies or socii. Although after the death of Nero in 70 AD they rose in revolt under the leadership of one Claudius Civilis3, the Batavi otherwise remained faithful to Rome. They became an important source of recruits for the Roman Army, many of them serving in the Praetorian guard, but also providing auxilliary forces who were particularly valued due to their specialised skills in the area of amphibious assaults.
In the sixteenth century under the influence of the ideas of Cornelis Aurelius it became fashionable to view the Batavi as the ancestors of the modern Dutch and in particular to see the revolt of 69 as a historical model for the contemporary revolt against Philip II of Spain. Hence from the tribal name comes Batavia, which was used as a synonym for Holland or the Netherlands particularly in respect of the short-lived Batavian Republic of 1795 to 1806.
NOTES
1 Some sources such as Brewer's Dictionary of Phrase and Fable state the tribe was Celtic, following the misattribution by Dio Cassius, but the consensus seems to Germanic, probably a sub-tribe of the Chatti.
2 The insula Batavorum itself was also to provide the Roman Empire with a valuable defensive position and a base for expeditions across the Rhine.
3 The rebellion against Roman rule was subdued within a year but Claudius Civilis was able to negotiate a favourable truce.
4 Sourced from German Historical Museum website see http://www.dhm.de/ausstellungen/mythen/english/index.html and the 1911 Encyclopedia Britannica