I'm sure a lot of people have these...or know someone who does....those little fat pockets right near your armpits so that when you're wearing a strapless dress or tanktop they kind of fold over and look nasty.
I try to go to the gym often but I just don't know what to do to target this area. I have very weak arms and bursitis in my right shoulder so I'm often afraid to strain it too hard although I know that I need to work it to keep it in shape.
Any tips on how to target this area [and maybe upper body without straining my shoulders]??

Well, it is a fat pocket, so while strength training will increase your lean muscle and by extention your metabolism, what you want to aim for is fat burning. Start with light weight training at least twice a week (full body exercises), cardio at least 30 minutes three times a week, and clean up the diet. As you get leaner, those tough spots will reduce as well.