Underclothes in the Middle Ages and Renaissance were constructed largely of rectangles. We see gussets in the traditional attire of India and the Middle East also. Being able to set one into a sleeve is a very useful technique. Shirts and smocks cut from rectangles benefit from the addition of an underarm gusset because they allow the wearers arm to have full range of motion without ripping the sleeve out. Such a garment also fits better. One can add an underarm gusset to choli also.
I cut this gusset from hanky weight Irish linen. It is 4 by 4 inches square. It can be cut straight on the grain, I would not cut it as a diamond (on the bias). If you are sewing for a child you can scale down to 3 by 3 inches. I would not enlarge this gusset for an adult size unless you are sewing for Shrek. Even then I would only go to 5 by 5 or the gusset will sag and bag.
Sew any point into the L shape where the sleeve meets the body of your garment. You can serge, but I like to French seam. Flip your work over and repeat. The fourth photo demonstrates completing the square.