There is nothing special under your arm. Most likely, it is your ribs. You may have broken or bruised ribs.
Most broken ribs heal on their own within six weeks. Restricting activities and icing the area regularly can help with healing and pain relief.
It's important to obtain adequate pain relief — if it hurts to breathe deeply, you may develop pneumonia. If oral medications don't help enough, your doctor might suggest injections of long-lasting anesthesia around the nerves that supply the ribs.
Once your pain is under control, your doctor might prescribe breathing exercises to help you breathe more deeply because shallow breathing can put you at risk of developing pneumonia.