The Island of Capri is located on the Amalfi coast opposite Sorrento and Positano in the Sea of Naples. I got there by ferry and then got into the upper town via the funicular train.
The name "capri" is a little confusing. First, there's no consensus on where the name came from. It could reference wild boars (κάπρος) if the name began with the Greeks or wild goats (capreae) if the name began with the Latins. If, on the other hand, it began with the Etruscans, then it could just mean "rocky." Whichever it is, the island is indeed rock and has goats and pigs on it...
The confusion doesn't stop there. The Island is called Capri. The main town on top of the island is also called Capri. The larger half of the island which is also a bit cheaper and rockier is called Anacapri.
For sure, the Island has been inhabited constantly since the times of the Romans. In fact, the whole area surrounding the Sea of Naples has been constantly populated despite the three local active volcanos. The area was where the super-rich of Rome had their summer palaces and acted as a kind of adult Disney-world with the strong emphasis on adult if you catch my meaning. The isle of Capri was where the Princeps Augustus built his summer palace and when he discovered artifacts dating the neolithic era. The Roman never found any native inhabitants, so at some point the original land owners got away or there was a previous volcanic eruption that took them out the picture.
After Augustus died, Tiberius built his summer home and became notorious when he refused to leave for ten years. Capri, then, was the de facto capital of the Roman Empire for the decade in which Jesus ministered, died and rose from the dead and in which the nascent Church was established. Tiberius' face was the one on the coin which Jesus showed the crowd and asked "Whose face is this and whose inscription?" (Mt 22:21)
After the Roman empire, the Kingdom of Naples claimed Capri and held it until Napoleon occupied it in 1806. Today, it's a part of the region of Campania in Italy.
The main point of interest for most folks on the Island is the view and the name. The Isle of Capri is synonymous would wealthy elites and the jet set.
The Blue Grotto) is an interesting cave formation which plays a trick with coloring the water and which is both popular and a tourist trap.