Depending on the breed, a dog is usually considered to be a senior between the ages of 7-9 while a cat reaches this point at 10 years. Medical problems including the incidence of liver, kidney, endocrine, & dental disease, cancer, and obesity greatly increase around this time.
Annual check-ups are extremely important in trying to catch these diseases early, so you can spend the longest time possible with your pet. Annual blood panels can be very helpful in assisting in early disease processes that can not be appreciated by physical exam alone.
It is also important to keep track of your pet's eating/drinking habits and their urination/defecation habits at home. Changes in these behaviors can help you see when something is wrong.
Feel free to ask your veterinarian any questions about the care of your senior pets.