It usually means the canned green chiles which are only a little bit warm. If you like more fire, use the jalapenos. If you want fresh, green chiles are not commonly available most places but anaheims are good and only a bit warmer than green chiles. Again, if you want more heat, use a fresh jalapeno
The exception to that is with Asian recipes. In those cases, they mean something with heat unless they call for a bell pepper. Jalapeno and Serrano are good choices there though there are plenty of other good choices too.
What's the difference? Firstly, heat. They're quite moderate in general though some varieties are certainly hotter.. Second flavor. New Mexico green chiles are a fairly specialty product. Purists order frozen chiles from New Mexico. They DO taste better than the canned in my experience. But these are thick walled chiles with a lot of meat to them to carry lots of flavor. They are also fairly thick skinned which makes them easy to peel when roasted.
This page will show you some of the differences. Look for NuMex in the type.Pepper Profile: New Mexican Varieties
The Big Jim is a LARGE pepper, but it's less controlled for heat as location, water and other variances play into how hot a chile is.