Dogs are neutered soon after they are fulling grown to reduce the risk of orthopedic problems in the future, if neutering is necessary. In toy breed dogs, that's usually around 6 months of age, while in giant breed dogs, at about 2 years of age. Ask your veterinarian when or if neutering is recommended for your dog.
There are minimal medical benefits to neutering a dog. Behaviorally, neutered dogs are less likely to urine mark and roam. Other behaviors are NOT affected. Neutering an aggressive dog will not make any difference in the dog's behavior. Aggressive dogs need behavioral modification not surgery.
Cats are neutered at 5-6 months of age. There are few medical benefits in neutering cats. Frankly most people neuter their pet cats because tom cat urine stinks- BAD. And unneutered male cats are highly likely to spray urine. Most veterinarians will agree- unless you plan to breed your pure-bred cat, get him neutered around 5-6 months of age.
We don't like to neuter cats younger than 5 months because it may predispose the cat to urethral blockages, a life-threatening condition in which they can not urinate.