Kids who grow up with dogs and pets learn empathy, responsibility and respect for nature. For many of us, helping care for the family dog is our first experience with chores and leaning to take care of others. A child who nurtures a dog, also learns to appreciate, not fear animals and nature and takes these important lessons into adulthood. In addition, your child will have a natural exercise partner. Having Sally and Tom running around and playing with a dog sure beats having them watch TV and gorge on junk food!
Moreover, kids who grow up with pets may have stronger immune systems and may be less likely to develop allergies, asthma and other problems. Here's another relevant article.
If you can't have a dog or cat, don't fret. Get creative and figure out ways to expose the kids in your life to animals. Although not as cuddly as a dog (or even a cat), a fish is a fine introduction to animals. Alternatively, take your kids to visit homes with pets and talk about the importance of respecting nature and being kind to animals.