Green chiles (with an "e" in NM) are indeed spicy (hot!) although how hot they are can vary a lot, depending on the variety of pepper, the amount of water it got during its growing season, when it was picked, etc. Chile is a staple of New Mexican cuisine (not to be confused with Mexican or TexMex cuisines), and it can accompany every meal, including breakfast. The peppers are roasted in the fall (around now -- you can small roasting chiles all over town), and can be served whole as toppings (e.g, green chile burgers or pizza) or in burritos or quesadillas, and is also ground up into a sauce that is put on anything and everything, but never served with beans and meat (as in "chili con carne" found in TexMex cuisine). You can also get red or green chile, and while green is usually hotter, that's not always the case (the red ones have been allowed to ripen before picking), and a restaurant will be able to tell you which is hotter that particular day. If you want both red and green, order "Christmas", and those new to chile would be advised to order it on the side(!).
New Mexicans often become addicted to their chile (due to the endorphin release it produces), and I know lots of folks who have it shipped to them, frozen, when they're away from NM. Chile peppers put Hatch, NM (a teeny town) on the map, as theirs are arguably considered the best in NM. Over the 10 yrs I lived there I slowly learned to eat mild or medium chile, but was never able to tolerate the hotter ones.