What is wrong with my baby have co on her chest coughing and a runny nose?

Bronchilitis. Sounds like it might be bronchiolitis which is typically caused by the RSV virus. Take him to your doctor asap a it can make breathing difficult.
See below. It's most likely a common cold. Cold and cough meds are proven not to help and can cause severe side effects. Use saline nose drops and suction the mucus from the nose as often as needed. Also, elevate the head of the crib.If your doctor ok's it, use a humidifier in the room during sleep. If these actions don't help enough, she needs to see her doctor.