Specially gourmet cooks go for copper when they have to cook foods at exact temperatures. Copper is the best choice in this case because there is no better conductor of heat. You can cook at exactly the temperature you want, e.g. you can cook at 78 degrees celsius if you wanted to. Great for frying and sauteing.
Downsides of copper are high price and the fact that copper reacts with food when heated and thus needs special coating (most of the time stainless steel).
Cast iron cookware is kind of a special case. It needs to be conditioned before using it because it will otherwise react with certain kinds of acidic foods and absorb the flavors of the food. (There’s a special kind of cast iron that doesn’t need conditioning - enameled cast iron cookware. A hard procelain enamel coating protects the cast iron from the food and vice versa.) The advantages of cast iron are that it provides even heating at high cooking temperatures and retains the heat excellently. One downside of this material though is that it heats up slowly.
Cast iron cookware is excellent for baking, browning and frying foods.
Anodized aluminum cookware has all the advantages of normal aluminum (except for the low price) but makes up for some of regular aluminums disadvantages. A chemical process makes aluminum much more durable and gives it a kind of non-stick surface and also prevents leakage of toxic substances into the food. It really is kind of the cheap version of stainless steel. Can be great for roasters, Dutch ovens, stockpots and saute pans.
However, you’ll have to handwash it because dishwashers might damage the surface. You also should not use steel wool or other aggressive scrubbing tools or corrosive detergents.
Another problem can be that because of it’s dark color it can be hard to observe the food - sometimes you’ll have to judge your food by it’s color and that might be difficult with anodized aluminum if you don’t have optimal lighting.
Glass cookware is a good choice if you use your microwave a lot. While most of the times it’s used for baking, there is also stovetop glass cookware available (it usually needs special handling though). Foods sticking to glass cookware is pretty unusual because glass is in itself already a non-stick material.
Disadvantages are that it’s heavy weight, and it’s heat distribution (not evenly) and that it’s not easy to handle. Glass is a very save cooking material, I have never heard of negative health effects from glass cookware.
Ceramic cookware has very similar characteristics to glass cookware (and can also be used for microwave cooking and is pretty non-sticky) but retains the heat better than usual glass cookware, distributes heat very evenly and is oftentimes also non-sticky. Oftentimes it’s used to make creme brulee, flan and custard.
Unglazed ceramics like terra cotta have are kind of spongy (that means they have lots of tiny little wholes inside) and thus can make the food a little juice, because the surface of the cookware add moisture in the form of steam to the food.
One thing that you have to be very cautious of when purchasing unglazed ceramic cookware is that the clay doesn’t contain lead.
Ceramic cookware might brake if you don’t handle it careful - but it’s also pretty cheap generally.