Green chili (chili with an 'i' at the end, as opposed to chile with an 'e' at the end) might be one of a couple things:
1) A misspelling of green chile, which is a word for a New Mexican dish as well as a word for the principal ingredient in both that dish and many other New Mexican dishes.
2) Some kind of chili recipe which uses ingredients that lend it a green color. These could include chili having as an ingredient actual green chile (which ingredient is correctly spelled with an 'e', even as the recipe is spelled with an 'i' at the end), or, as is more likely the case with people who use the 'chili' spelling, having green tomatillo as an ingredient.
When people refer to "New Mexico green chili" they more often than not are making a spelling error. Or, worse, they mean tomatillo and they intend to inflict something on consumers that has nothing to do with New Mexico, and they are using "New Mexican" simply as a marketing buzzword, without understanding what they are promising.
And if they refer instead to simply "green chili" (chili with an 'i' at the end) and mean to indicate something that fits option number two above, a pot of chili having ingredients that make it green, then they are not talking about anything that is remotely a New Mexican dish.
We should clarify the distinction between "green chili" made with some green chile in it, and the dish called green chile, made primarily of green chile:
The dish called green chile consists of cooked green chile, often with onions and garlic, and can be either a sauce to be added into (not just onto) other food dishes, or a dish itself, as in a "bowl of green", in which case it sometimes also has beans and meat added. That last version, in the bowl, sounds a bit like what people mean when they say "chili" but the difference is that with chili, the beans and meat are a lot heavier and dominant. With the dish called green chile, the ingredient green chile is dominant, and beans and meat, if any, are a small part.
Finally we must also remove any confusion between a "bowl of green" as described above, and an entirely different dish in a bowl, Green Chile Stew, which is also properly made with green chile as an ingredient.