There are many diseases out in the community that your pet can be protected against through vaccination.
With puppies the most common and serious disease concern is parvo. This is a devastating disease for a pet owner and puppy to go through of which there is no cure. Symptoms include severe lethargy, vomiting, and diarrhea which is often bloody. Certain breeds tend to develop more serious signs and many pets do not survive even with extensive medical care. Puppies should have a minimum of 2 vaccinations in the puppy series before visiting high traffic dog areas.
Distemper is also a very real concern that does appear on occasion within young dogs. Symptoms usually appear initially as what would appear to be a cough progressing to nasal/ocular discharge. Neurological signs can then develop which can vary in severity from muscle tremors to actual seizures. These nervous symptoms once developed are permanent. There is no cure for this disease progression.
Kennel cough is also quite prevalent in areas where large groups of dogs tend to congregate, with a deep husky cough being the most obvious sign. Although tending to be viral in nature, antibiotics maybe warranted for those dogs with more severe symptoms.
Other vaccines that your veterinarian may recommend for your dog depending on your lifestyle and travel habits would be Rabies, Giardia, and Lyme.
Regarding vaccinations available for cats, the most important would be for the group of diseases with upper respiratory symptoms. Rhinotracheitis and Calici-virus are the two most prevalent. Signs usually seen are sneezing, ocular and/or nasal discharge with more severe symptoms ranging from lethargy and fever. Animals should be watching closely for worsening symptoms and to ensure eating. Checking in with your veterinarians is strongly advised.
Other vaccines to consider for your pet would be Panleukopenia, Feline Leukemia, and Rabies.