The Short Answer:
Cats can have the juicy part of the peach, same as humans, but they cannot eat the entire peach. If you are going to give your cat some peach, then you should peel it, and remove the pit. The skin can have dangerous pesticides that affect cats more than humans (because of their size) and the pits are both a choking hazard, and have a poisonous center.
If you are giving your cat some peach you should not give it a large amount. Although peaches are fine in small amounts, a large amount will give your car an upset stomach.
The First Time:
If you are giving a peach for the first time, you should note that some cars, like humans, are allergic to peaches. You should start by giving your cat a small amount, and watch him or her for a few minutes. If he or she shows signs of a severe allergic reaction, you should find an emergency vet. Otherwise you can give your cat peaches without worrying.