Welcome Erika,
Not sure exactly what your questions are, but I have been in a similar state of confusion when I was first introduced to this stuff.
It is an address to locate a parcel of land. Each parcel of land in the west was gridded out in parcels of land measuring one mile by one mile “sections”, which corresponds to 640 acres. These are numbered 1 though 36 to form what is called a “township.” You can read more about it here: Section (United States land surveying) - Wikipedia The U.S. Bureau of Land Management maintains a web site that allows you to find the original patent (the first deed) and, often, subsequent owners. You simply put in the address: Search - BLM GLO Records
I don’t own any rights in NM, so I cannot speak to what that state might also have. My parcels are in ND and that state maintains a website that allows owners to research drilling activity and other details.
You can just search using the details you have - when I searched in my browser I immediately found there is a Mineral Rights Forum on this very section:
Section 31 21s 27e Eddy County
Hope this helps get you started,
Greg