Cats are mostly strict meat eaters, however some choose to eat grass if they are sick and need to puke (horses usually cannot). You probably shouldn't let your cat eat grass, my cat eats it for fun sometimes as im not entirely sure. Trust me, my cat ate grass once and it was not pleasant at all. Cats are domestic, however their wild ancestors ate only meat, so cats are not just purely domestic, my cat still is wild. (I regret letting her near a dog, it wasn't fun to hear)
My cat eats a variety of foods, usually meow mix or special kitty (it's great food!), but she eats kit kaboodle now. (my mom picked that out for her)