The first procurator of Judaea. He was sent to Judaea, presumably by
Augustus Caesar, at the same time as a man named Quirinius (also called
Cyrenius), to make an account of the wealth of the province, and "to
dispose of Archelaus's [the former Herodian king] money." It seems that
Coponius performed the function of procurator of Judaea, while Quirinius
was sent to be the Legate of Syria, though both worked closely together
on the census.
It is also this census that is mentioned in Luke 2:1-2: "In those days, a decree went out from Caesar Augustus that all the world should be enrolled. That was the first census, when Quirinius was governor of Syria." The only problem is that this statement in Luke it is about 9 years off. If Christ was born in 4 B.C., as is commonly understood, and this census was the reason why Joseph and Mary had to travel to Bethlehem (and thus why Jesus was born in a manger), and the census was not taken until Quirinius got to Palestine in A.D. 6, then something is amiss somewhere. Perhaps the best answer that I have seen is that Luke may have meant "This was the first census before Quirinius was governor."
While Coponius was governing the province, Josephus tells us about the following incident: The priests of Jeruslem's Second Temple kept the gates to the Temple closed after midnight because a group of Samaritans (residents of a province to the north of Judaea who are often mentioned in the Bible and were generally disliked by the Judaeans) had come to the temple at night and threw human bones (or perhaps corpses) around the building (Antiquities 18.2.2).
He also writes of a mover and shaker in the province named Judas who urged his followers not to pay taxes to the Roman government, because to do so, he said, was to recognize mortals as their lords which would, he claimed, deny God's soverignty (War 2.8.1). What interaction Coponius had with this rebel is lost to us.
Ancient sources: Antiquities 18.1.1, 18.2.2; War 2.8.1.