Cats are obligate carnivores. That means they must eat animals; taurine in particular is an essential part of their diet and can not be found in plants.
Since they eat organs we don't (including the digestive system and bones), they get the full benefit of all the vitamins and minerals the prey needed to keep itself alive. That is a pretty darned complete diet for a cat. Not scientifically perfect, but most of us don't eat a perfectly balanced diet either; it just means they need to eat a bit more to cover their needs.
Cats will sometimes eat plants. But they can't taste sugars, and they don't have all the liver enzymes we herbivores and omnivores have to handle the toxins plants produce to discourage critters from eating them, so most plants are more interesting for cleaning teeth or to prevent hairballs than as food.
But there are exceptions, on a per-cat basis, especially when it's a food their human eats. I've known cats that liked broccoli,. I've known one who was absolutely wild about white bread (but not the crusts). My own little old lady wants to sample my graham crackers; I think she likes the texture, since it isn't that far from her dry food. Decades ago, a friend's cat would sit under the high chair to grab any cheereos the toddler dropped. Cats is weird little alien minds...
But to get back to the question, plants really aren't part of the normal feline diet, and a cat can't be a vegetarian without serious dietary supplements (and probably would not appreciate your trying to convince them). Carnivore means meat eater, but for a broader definition of meat than just muscle. This doesn't mean that cat foods which contain plants are evil, just that they are carefully balanced by the manufacturers to meet the cat's well-known needs, which may indeed involve adding supplements -- just as many packaged foods for humans do.
