Momo is a category, not a dish. The same filling and the same wrapper can arrive at your table in six quite different states, and knowing which is which saves you from ordering the wrong one on the wrong day.
Steamed is the original and the benchmark — bamboo basket, ten minutes, served with achar. Kothey is steamed first and then pan-fried on one face, which gives you a crisp base and a soft crown. Fried goes into oil from raw and comes out closer to a pastry; it travels the best and holds heat the longest.
Jhol sits in a warm, thin, sesame-heavy broth and is the one to order when it is cold outside. Chili momo is fried and then tossed through a wok sauce with onion and capsicum, which makes it a plate to share rather than a starter. Open-topped momo skips the crown entirely and is finished with whatever the kitchen is proud of that week.
Where they come from is a longer story than most menus admit. The fold travelled from Tibet into the Kathmandu Valley and settled with the Newar community, which is where the tomato-and-sesame achar enters the picture. What is served in Nepal today is not what was served in Lhasa, and neither is quite what we serve in New York — every kitchen the recipe passes through leaves something in it.
A rough rule if you cannot decide: steamed in spring, jhol in winter, kothey when you are hungry, chili when there are four of you.