It is called a gusset. A gore would go along a hemline and a gusset is inserted into a seam. The following link is for proper definitions.
http://www.renaissancetailor.com/demos_goresgussets.htmLook at the armpit seam of a shirt. It looks like an x or lower case t. If you open the seam in all 4 directions from that intersection by about 2-3 inches you will notice that the hole you made looks like a diamond or square shape. All you need to do is take a diamond shaped piece of fabric that is bigger than the hole by 1/4 of an inch in all directions and sew it into the hole. As with any garb, make sure you are sewing on the inside of the garment so that your raw edges end up inside.
The following link takes you to a tutorial about gussets. It is a little strange since it's the seam on pants but the same goes for the seam on armpits. The most important images here are the last 2 pictures that show the end result. The trick to making it work for you is to add the gussets before you have a problem. They will do very little once you have ripped the garment but you can always sew it up and patch it and add the gusset and hope for the best on your already ripped item. Good luck.
http://www.fishsticksandfries.com/Little%20One%20Layette/Gusset%20Tutorial.pdf