In Ancient Rome the genii were the spirits that protected specific people and places. Some worked one to a person, much like guardian angels, while others worked more like patron saints, watching over specific areas of life. Almost anything, including families, houses, doors, gates, streets, hills, districts, villages, and tribes, could have a genius associated with it. (Genii is plural, genius is the singular). Likewise, almost any concept could have genii too; for example, the genius of marriage was Iugalis, the genius of sleep was Cuba.
The Greek equivalent were the daimons.
A genii is not the same as a jinni.
It is also worth noting that genii is the correct Latin plural for genius (although 'geniuses' is more common in modern English). A Latin word ending in -us becomes plural by exchanging the -us into an -i. Since genius already has an 'i' before its -us, we get the -ii combo. A few other words share this trait, such as radii (plural of 'radius')and gladii (plural of 'gladius').