Dark Obelisks are stationary devices found in various tombs in Egypt (Act II) and some caves in the Orient (Act III). They look like small pillars with a glowing blue-green skull on their top and are capable of creating several Hideous Phasmas, fast-moving blue-green glowing skeletons (undead).
A Dark Obelisk will create a total of four Hideous Phasmas - first three melee fighters and then one slightly more powerful ranged fighter. If a Phasma is killed, it is immediately replaced by the Obelisk. If the Obelisk is destroyed, all Phasmas it created will vanish along with it.
Usually it's the best strategy to ignore the Phasmas and directly attack and destroy the Obelisk, thereby also killing the Phasmas. They won't do that much damage anyway (except for the ranged-attacking Phasma, but it's just one).
Note: The Canopic Shrines found in the boss chamber in The Great Sphinx share the same appearance with Dark Obelisks and fulfill a similar function.