An Indian kadai and an American skillet both sit on a burner and both cook, but they behave differently with the same ingredients. The geometry is doing the work.
Curved walls pool a small amount of oil
A skillet spreads any volume of oil into a thin film across a wide flat base. Getting real depth for shallow frying means using far more oil than the food needs.
A kadai collects the same oil into a narrow well at the bottom. A modest quantity becomes deep enough to submerge pakoras or puris, which is the frying depth Indian cooking assumes.
That matters beyond economy. Oil at depth holds its temperature when cold food goes in, while a thin film in a wide pan drops sharply and produces greasy rather than crisp results.
Heat is graded from base to rim
The base sits directly over the flame and runs hot; the sloping sides run progressively cooler. That gradient is a working surface, not a defect.
Ingredients that have finished can be pushed up the wall to hold while something else sears at the bottom. A flat pan offers no equivalent parking space at a different temperature.
Reduction happens upward, not outward
A gravy in a skillet spreads thin and loses water fast, which sounds useful until the sauce reduces past the point where the solids in it are still cooking evenly.
In a kadai the same volume stands deeper with less surface exposed. Reduction is slower and more controllable, and the sauce can bubble hard without spitting across the stove.
Deep walls also allow vigorous stirring. Tossing a dry sabzi in a skillet throws food onto the burner; the same motion in a kadai returns it to the center.
Material changes the response
Cast iron and carbon steel kadais hold heat and are slow to recover, which suits long-cooked dishes. Thin aluminium responds instantly and suits quick frying where temperature is adjusted constantly.
Nonstick versions exist but limit the high dry heat that many preparations depend on. Uncoated metal is the traditional choice for a reason.
Buying one in the United States
Indian grocery stores stock them, and the practical decision is diameter against burner size. A kadai much wider than the burner heats only in the middle and defeats the gradient entirely.
Flat-bottomed kadais exist specifically for electric and glass-top ranges, where a fully rounded base makes poor contact. A round-bottomed one on a coil or induction surface will heat slowly and unevenly.