In India, curry is traditionally cooked in a deep pot with straight sides called a degchi. Curry is, simply speaking, a dish with a sauce, made with several ingredients like spices, herbs and aromatics and bulk ingredients like fish, meat or vegetables, with a liquid making the sauce.
Curries differ widely between the regions of Asia because of differences in, for example, the climate, the terrain and historical influences. In the north, for instance, yoghurt and cream are used in the sauces whereas in the south the cook will more often use coconut milk. Some sauces may contain one solitary spice where others contain a blend of as many as twenty.
Most curries can be made hotter or milder by adding or omitting ingredients. Some of the mildest, and creamy curries will never be made hotter because that would spoil them. There are hundreds of differently named curries and also many variations in the spelling of many, because the Asian restaurant owner (or his printer) has translated from the Asian names and written what he thought was the best spelling.
In the following answers I will list some that I know and will categorise them as they are most popularly served. Please bear in mind that this will not always be precise.