It is not a good idea to let cats eat scraps off plates for two reasons.
It encourages bad behavior. It would be better to select any suitable scraps and put them in the cat's food bowl. This also ensures that what the cat is eating is being supervised. When selecting table scraps for you cat, pick off cooked, unseasoned meat and feed it to your cat in her food bowl.
There are many things that can be, potentially, harmful for a cat to eat from human leftovers.
Sometimes cats will vomit up table scraps, as they don't cope well with big diet changes, they cope better with gradual changes and introduction of new foods. Also the type of seasonings and spices human foods have frequently do not agree with a cat's gut and can cause irritation.
The following list is by no means comprehensive
Avocado contains persin, which is a dangerous toxin for cats. It can cause gastrointestinal upset, cardiac edema and difficulty breathing. For more detail see here https://pets.stackexchange.com/questions/939/is-it-safe-to-feed-my-pet-avocado
Broccoli contains isothiocyanate which if given in greater than 10% of the total diet will cause gastrointestinal upsets, if it reaches 25% of the total diet, can (and most likely will be) fatal.
Cooked bones are not recommended to feed cats or dogs. Cooked chicken bones tend to splinter and can cause damage to the gut by perforating the bowel or causing a blockage, as they can be hard to pass through the bowel.
Grapes and their derivatives are also bad for cats. This is discussed in more detail here Is it safe for my dog or cat to eat grapes, raisins and sultanas?. Grapes can permanently damage a cat's kidneys and can cause renal (kidney) failure in cats.
Liver can cause vitamin A toxicity.
Milk, cats are do not produce lactase and cannot digest cow's milk. It can cause diarrhea. Addressed in more detail here Should I feed my cat milk?.
It's not advisable to feed your cat nuts. Nuts are a more complex issue addressed here https://pets.stackexchange.com/questions/940/is-it-healthy-to-feed-my-dog-or-cat-nuts.
Onions and plants from the onion family, like garlic are addressed in detail here Are onions dangerous for my cat or dog and why?. To give a brief overview, onions and garlic are frequently used in cooking and are also found in many processed foods, even baby food. Onions cause hemolysis of red blood cells (RBC). It causes the RBC to break, which can lead to a potentially fatal condition allium toxicosis.
Sweets left on plates are also not a good idea to feed cats. If they have a taste for some leftovers using Xylitol can cause hypoglycemia in cats and can lead to coma and be fatal. This is discussed in more detail here Is it OK to feed my dog or cat candy with sugar substitutes?.
As for sugar, generally, it serves no good nutritional value for a cat, it has a high glycaemic index, which can affect blood sugar levels and over time increase a cat's chance of developing diabetes, not to mention tooth decay.
There are also risk in feeding your cat some other vegetables. For more detail see here https://pets.stackexchange.com/questions/941/is-it-ok-to-feed-my-dog-and-cat-vegetables