The find
In June 1961 an Italian expedition led by Antonio Frova was excavating the theatre at Caesarea Maritima, Herod's harbour city and the seat of Roman government in Judaea. A limestone block roughly eighty by sixty-five centimetres was found built into a stairway as a landing. It carried an inscription, worn and broken at the edges, and it had plainly been cut down from something larger before being reused - probably in the fourth century, when the theatre was remodelled and older ruins were quarried for material.
What survives
Enough. The readable portions preserve the name Pilatus with the end of Pontius before it, and the title praefectus of Judaea. The fuller reconstruction - that the block came from the dedication of a building, probably a temple, raised in honour of the emperor Tiberius - depends on restoring damaged letters and is presented by scholars as reconstruction. The name and the office do not.
Why the title is the interesting part
Roman governors of Judaea held the rank of prefect until about AD 44, after which the province was administered by procurators. Tacitus, writing early in the second century, calls Pilate a procurator - the term of his own day, applied backwards. The stone gives the contemporary title, prefect, which is what a man in that post in the AD 30s would actually have been called. It is a small correction, and it is the kind that only a contemporary inscription can supply.
Where it sits in Scripture
Pilate appears in all four gospels at the trial of Jesus, and Luke dates the beginning of John's ministry by his governorship. Caesarea, where the stone was found, is where Pilate would ordinarily have lived - Jerusalem was the place a prefect went for the festivals, when crowds made trouble likely. Acts has Paul sent down to Caesarea under guard and held there for two years, before the same seat of government a generation later.