Capacity, base and pouring spout
The capacity determines how much you can make at once. A mixing glass of 55 to 65 cl is good for one, at most two cocktails, because alongside the drink you need room for a decent amount of ice. From 70 cl upwards you make three at a time, and that is the size most used behind a bar. The models of around 80 cl, sometimes footed, are meant for working ahead.
After that, look at the base and the rim. A heavy, thick base keeps the glass stable when you stir with force and does not draw the cold away immediately. A proper pouring spout is no luxury: without one, the drink runs down the outside as soon as you strain. The yarai pattern, that diagonal diamond cut you see on many models, is Japanese in origin and gives grip to wet hands. If your main concern is keeping the glass cold, a double-walled model is the logical choice.