Puppies breathe faster and more erratically when they are at rest or sleeping. Their heartbeats are also faster than those of an adult dog. This is completely normal and natural. Puppies' rapid breaths and heartbeats will slow down as the body matures and develops.
Puppies may even appear to forget to breathe during sleep. Their breaths may seem to stop and start at arbitrary times, or they may come in short, ragged, shallow bursts. If this occurs solely when a puppy is in a relaxed or sleeping state, it is nothing to be concerned about medically. Sometimes, it can even be a result of the puppy's dream state. If they are dreaming about something exciting, their heart rate may increase.
If, however, a puppy's breath is fast, shallow, or otherwise abnormal while awake and active throughout the day, it may be a cause for alarm. Occasionally, dogs are born with heart defects or circulatory problems which may cause them difficulty breathing. If the puppy's breathing problems are combined with a persistent barking cough, it could be a sign of kennel cough, which requires medical attention. If the puppy's stomach is swollen unnaturally, it could be a sign of worms.
A healthy dog's heart beats at a rate of 60-160 beats per minute. Consult a veterinarian if there is any reason for you to believe that your puppy's breathing problems are abnormal and potentially dangerous.