A cattery is a place where cats are boarded and/or bred.
In general, there are two types of cattery.
A boarding cattery is a place where cats are temporarily housed while their owners are unable to care for them, whether due to illness, travel, or home renovation.
A breeding cattery is a place where a cat breeder breeds cats, generally for commercial sale.
Either way, you have to admit "cattery" is a pretty funny word.