The popular picture has to go first, because everything else is misread as long as it stands. In that picture the Christians of Rome fled underground and lived there, holding secret services by torchlight while soldiers hunted them in the streets above. It is a wonderful image, and in its strong form it is false. It comes largely from the nineteenth century, from devotional novels like Wiseman's Fabiola and Sienkiewicz's Quo Vadis, and that fiction has outlived the scholarship by a century and a half.
What the catacombs actually were is simpler. They were cemeteries. Roman law forbade burial inside the city walls, a prohibition reaching back to the Twelve Tables, so every Roman burial ground, pagan and Jewish and Christian alike, lay outside the walls along the consular roads. The rock beneath the Roman campagna is tufa, a volcanic ash soft enough to cut with a hand pick and stable enough that the galleries do not collapse. Christians cut narrow corridors, lined the walls with graves, and dug down to a new level when one filled. None of this was secret; the entrances stood in the open on well-traveled roads. Nor was it lawless. A Roman tomb was a locus religiosus, and violating one was an offense regardless of whose body lay in it. Roman law protected graves even when it did not protect the living who dug them.
And here the legend, having been dismissed, must be partly let back in. Christians did use the cemeteries for assembly, and the government knew it. Valerian's first edict in 257 forbade Christians to hold assemblies or to enter their cemeteries, and we know it from the acts of the proconsular trial of Cyprian of Carthage, where the proconsul states the prohibition to his face. On the sixth of August, 258, Pope Sixtus II was seized and killed in a cemetery on the Appian Way while seated teaching the congregation, and the source is not legend but a letter of Cyprian written within weeks. Two years later Gallienus reversed course, and Eusebius preserves the rescript restoring to the bishops the places of the cemeteries. Diocletian confiscated them again in 303.
So the accurate statement is narrower than the legend and more interesting. Christians did not live underground; there was no water, no food, and no hiding a congregation for years in a place the authorities could find on a map. But they gathered at the graves of their dead, knowing it was forbidden, and some of them were killed there.
The word is an accident. Ad catacumbas was a local name for a dip in the ground near the third milestone of the Appian Way, usually explained from the Greek kata kymbas, roughly "down by the hollows." The Christians' own word was coemeterium, from the Greek for a sleeping place. That is where the English word cemetery comes from, and it was a theological claim.