Shortness of breath is usually nothing to be concerned about. Unless it is due to asthma, it will dissipate with more training.
Shortness of breath while running can be caused by lack of conditioning, running at too fast a pace, or exercise-induced asthma.
Lack of Conditioning
Shortness of breath in a new or returning runner most likely is due to a simple lack of conditioning. Start out slowly and build up speed and distance.
Exercise Induced Asthma
If your shortness of breath is accompanied by mucous production and tightness in the chest, you may have exercise-induced asthma. There are many medications that can control this condition.
Running at Too Fast of a Pace
Even if you are a conditioned runner you may simply be attempting to run at too fast a pace. Run at a pace at which you could talk with another runner next to you.
If your shortness of breath is extreme or painful, stop running and seek medical attention immediately. You could be suffering from a serious condition.