There is a purpose to every shape
You must know that there is a specific shape for every cooking chore. You should choose the shape of your cooking vessel in dependence from the meal you intend to prepare and the cooking technique you are going to use.
For example, sauce pans have straight sides to contain the added liquids during cooking while a fry pan has sloped sides to facilitate the flipping and turning of the food.
Fry Pan: The sloped sides facilitate flipping and turning, as well as releasing food from the pan once you are finished. It’s ideal for fast cooking at higher heat. This is usually one of the most used pieces of cookware in every kitchen.
Stockpot: a large and deep vessel with a flat bottom. The tall profile of the sides makes it ideal for boiling and steaming, as well for preparing soups and stocks. It’s great for boiling pasta because of the tall sides.
Sauce Pan: this is a more versatile vessel. You can use it to make sauces, vegetables, grains, soups and cereals. The sides are straight and you can use it with or without the lid to control evaporation.
Saucier: has sloped sides to enable whisking and stirring. This pan is ideal for building sauces and dishes that require continuous stirring (like risottos).
Dutch Oven: the Dutch Oven is usually a larger vessel. It can be round or oval shaped. The Dutch oven is a very versatile cooking vessel, but it’s ideal for slow cooking of stews, pot roasts and braised meat. The modern version comes with the handles to make lifting easier (because these vessels are usually quite heavy).
Braiser Pan: this vessel is designed for oven slow-cooking and stovetop browning. You can use it for serving as well. The tight-fitting lid retains the moisture, so this piece is ideal for making juicy and tender meats.
Saute Pan: similar to a frying pan, the difference being that a saute pan has high and straight sides (not sloped) so more food can be prepared in it. The high sides help retain the liquids you add during cooking. It’s great for frying, searing and sautéing.
Windsor pan: has a narrow bottom and flared sides in order to facilitate evaporation of the liquids. The wide mouth helps precise pouring. This pot is ideal for reducing sauces.
Butter Warmer: it’s was made originally for melting and clarifying butter. It can be used for other purposes as well, like warming small amounts of liquids or even for making smaller side dished.