Perhaps I have a taste for the macabre, but this is one of my favorite Alexandria outings. Part Lord of the Rings, part Indiana Jones, the catacombs are about 35m (115 ft.) below ground level and reached by a spiral staircase that circles an open central shaft that was once used to lower dead bodies into the tombs below. The story goes that the whole complex was discovered in 1900 when a donkey, working the land above, fell in.
At the bottom of the stairs, you'll find yourself in a funerary complex, with rooms and passages leading off in all directions. It was more formally laid out when it was built in the 2nd century A.D., and many of the interconnecting passages that you see now are the work of grave robbers.
The burial tomb chamber is obvious once you're at the bottom of the stairs; it's through the doorway that's flanked by Anubis and Agathodaemon. Stop for a moment before going in to consider the odd mix of Egyptian, Roman, and Greek symbolism here -- it's an excellent illustration of the syncretism of Ptolemaic culture. Anubis is the Egyptian god most closely associated with tombs, and he features prominently on royal tombs in Upper Egypt, where he protected the mummified remains of the occupants. But here, instead of the traditional collar around his neck, Anubis wears the uniform of a Roman legionary. The snake-tailed Agathodaemon, meanwhile, is an expat Greek god associated with good food and plenty, which may or may not be a reference to funeral rites that included feasting in the tombs. The chamber is modeled on a temple, with an antechamber and an inner sanctum. Note: Even though the statues and carvings have been left unprotected and unsupervised, you shouldn't touch them.
The rest of the catacombs, though less elaborate, are well worth a visit. If you've brought water or a snack, try the triclinium, the large room with the benches close to the bottom of the stairs. This was where friends and family of the deceased gathered periodically to feast and pay their respects.
The amount of the water in the lower tombs goes up and down, and there are always limits to how far you can explore. Much of what you can access was added well after the original construction, and access was further improved by later raiders in search of treasure. There is a usually accessible section behind the main tomb where a narrow hallway goes off into a series of little tombs and a wholly separate section known as the Hall of Caracalla off to the right (standing with your back to the main stairs).
Note: Officially you are not allowed to take pictures in the catacombs, so tuck your camera into the depths of your bag (no X-ray machine here) and be discreet.