1- The chest:
*Musculoskeletal pain: including costochondritis (inflammation affecting the points where the ribs insert into the chest bone - associated with pain upon pressing against those points).
*Pleurisy (or pleuritis): inflammation of the pleura, the lining surrounding the lungs.
2- The heart: if you have risk factors for heart disease (obesity, smoking, high cholesterol, high blood pressure, family history of coronary artery disease), this pain can be classified as angina (chest pain of cardiac origin), which could occur with effort and is relieved by rest, or could even occur at rest.
3- The esophagus (the swallowing tube that connects the throat to the stomach): in this case, the symptoms fall under an entity known as dysphagia (difficulty swallowing, +/- odynophagia, or pain upon swallowing): could be secondary to a neurological problem, a muscular disorder, or disease of the esophagus that causes impairment in the motility or mechanical obstruction of the esophagus (whether from inside the esophagus itself, or something from outside impinging on it). Commonly, it is due to esophagitis (inflammation of the esophagus induced by acid refluxing from the stomach, usually associated with heartburn and indigestion). Specific concerns include whether difficulty swallowing includes solids, liquids, or both; whether food comes out the nose; whether there is drool or food spills from the mouth; and whether cough or choke occur while eating.
I urge you to seek medical attention to rule out serious underlying causes (like heart disease), and get to the bottom of it.