The hover robot is a guard or patrol robot, most likely designed before the Great War. It's very similar to the floating eye or the eyebot robot but far more heavily armed. After the War, it was used by the Calculator, the mad A.I. from Vault 0.
It uses a very quiet propulsion system to stealthily approach its target. It's armed with a rocket launcher and a very deadly melee attack, which it will use when it has exhausted its small supply of rockets. After it uses its melee attack, it will immediately cease functioning and fall to the ground.