Serious Eats has run numerous articles about the effectiveness of materials for various cooking purposes; in particular, woks and baking pizzas. I find their analysis usually very close to correct, but not quite. Thus, I am going to write a somewhat technical piece on my understanding.
Now there are three types of typical heating: convection, conduction and radiation. The former is mostly irrelevant for cooking. Convection is the transfer of energy by fluid flow, but in most cooking applications there is very little exchange of fluids. The exception might be deep frying as hot oil penetrates and cold water leaves, but I don’t yet know enough about deep frying. Convection ovens are technically speeding up conduction because they make sure there is hot air continually close to the food.
That leaves conduction and radiation as the major sources for cooking your food. Lets tackle the first. Obviously the conductivity of your cooking material is important for conduction, as the name would imply. However, how it factors in appears a bit vague to most people. It’s effect is better encapsulated by the thermal diffusivity and the thermal effusivity.
The thermal diffusivity measures how quickly a volume of material will reach thermal equilibrium or in other words how fast the entire thing will reach the same cooking temperature. The thermal diffusivity is proportional to conductivity, because we expect energy to be transmitted throughout the material quickly with higher conductivity, and inversely proportional to the volumetric heat capacity (the amount of energy needed to increase the temperature of a unit volume of the material) because higher heat capacity means it takes more energy to heat up. However, this term is relatively trivial for most cooking if you are preheating your pan. It just affects how long you have to wait. The exception here is probably baked goods where the vessel containing your goods starts at room temperature with your food. Conveniently, we usually bake in aluminum which has a relatively high diffusivity.
Conductivity also enters into the thermal effusivity. This is a measure of how easily a material exchanges energy with another material with which it is in contact, i.e. our food and the cooking surface. It is proportional to conductivity and volumetric heat capacity. The typical example is touching metal surfaces and how they “feel” cold. Your body sense energy fluxes or transfers, not temperatures, and so the high effusivity of metal, it’s ability to leech energy from your skin, makes it feel cold. Metals have high effusivity because they conduct well, obviously, but also because they have high heat capacity which means they can suck or spew out a lot of energy without changing temperature much and the difference in temperature of two objects is also a major factor in energy transfer.
When it comes to food what we need to do is compare the effusivity of the food to that of our cooking surface. This can then be used to compute the temperature at the surface of contact. At that point the problem is governed by the heat equation, an equation that basically tells you how the bulk of a material heats up if one side is at a given temperature and is based on the conductivity of the food, not your cooking surface. The thing is, the effusivity of your typical cooking surface vastly exceeds that of your food. Water probably contributes most to the effusivity of your food and even it has an effusivity 20 times lower than your typical cooking metal such as aluminum. Thus to a first approximation the contact surface is the temperature of your cooking surface, regardless of your material.
So did I write all that out to say that your cooking material does not matter? Well I am ignoring one thing which is the actual heat capacity of your cooking surface. The above is only true if adding food to your surface does not appreciably alter the temperature of said surface. This is why people use hefty cast iron because it has a high thermal mass so once you preheat it the addition of food barely lowers its temperature. So cast iron is best, yes? Actually, if you look at the volumetric heat capacity, the product of the specific heat capacity and the density, all of your typical cooking materials are about the same. It’s just that for some reason they make cast iron pots and pans much thicker than pots and pans of other materials. The outcast here is actually aluminum. It has twice the specific heat, but a third of the density. This means you need a pan that is 1.5 times as thick as other materials to get the same heat capacity. The flip side is that said pan will actually weigh half as much as the pans made of other materials. Thus, in the ideal world we would all be using thick, but light, aluminum cookware.
Finally, a quick word on radiative heating. This is actually a very important source of heating that seems highly neglected. Your broiler mostly cooks via radiation and we know how quickly they can work. Anyone that has done a lot of baking can tell you that dark pans bake goods significantly faster than light pans. This is because dark pans absorb radiation easily, thus heating up faster, and emit more radiation at a given temperature than a light surface. Usually this is given by the emissivity or absorptivity which quantifies how much the material is like a true black body that absorbs all incoming radiation, with a factor of 1 meaning it is a true black body. Radiative heating can warm the interior of food faster than conductive heating, which must start from the contact surface, because it can actually penetrate the surface before being absorbed. Unfortunately, information is not readily available for me actually quantify the approximate ratio of conductive to radiative heating when cooking food. In terms of cooking technique, tasty reactions like the Maillard reaction that occurs when browning food is due to conduction whereas radiation cooks like a microwave. Still, if you want to cook more quickly, then go with dark cookware. If you want more even heating, such as with baking, a light pan will make sure the exterior does not burn before the interior is finished.
A few other considerations are needed as well. One, thick aluminum is not readily available despite being a relatively cheap material. Secondly, aluminum is highly reactive and should thus be clad in some other material of lower reactivity. Lastly, sometimes you have specialized needs that aren’t covered by the criteria above. For instance, maybe you would rather have a wok that reacts much more quickly to changes in heat (like taking the wok off the burner for a moment), then you would not want a high heat capacity wok.