Cats are symbolic of rebirth and resurrection, per their nine lives. Because they are nocturnal, they are also associated with darkness. Darkness often goes with fear, the unconscious, and things that are hidden. Cats are also symbols of mystery and magic, as aforementioned, but also unpredictability and even healing.
Cats have been both feared and loved throughout history by many different cultures. Some Native Americans, for instance the Oglala, would not have anything to do with any felines because they believed that cats had powerful magic and the ability to curse people. However, the Egyptians had great reverence for cats. They had multiple cat goddesses, including Bast, Sekmet (goddess with the head of a lioness), and Mafdet (goddess having come before Bast, who represented feral Egyptian cats). Each of these gods is female, which supports the association many have between cats and women. Cats are also associated with female medicine.