The definition of a zoonotic disease is one that can pass from animals to humans. The list is long ranging from Rabies at the top end of severity to Ringworm at the lower end of the scale. There are bacterial infections such as E Coli and Salmonella. Internal parasites such as Toxoplasmosis which is particularly problematic in pregnant women. Skin infestations such as Sarcoptic (fox) mange which will infest people and many many more.
The list is frightening but we have been living with animals for millennia and actually if we exercise common sense hygiene, washing our hands regularly after handing animals and especially before preparing or eating food the risks are minimal.
Toxoplasmosis deserves special mention regarding pregnant women. It is mostly caught via handling of cat faeces and will effect foetal development. Therefore pregnant women should exercise very high levels of hygiene and wear gloves if they have to handle cat litter trays. Avoidance of such contact would be preferable.