The NM rule book has come a long way over the past 10 years, but here's a primer on NM licensing. Obviously there's lots more details, and I'm only touching on a small part of the system.
Hunters generally acquire licenses to hunt in NM (most to least common) 1) by winning in the draw, or 2) after purcasing a "land-owner authorization" typically sold by ranchers which allows one to buy a license from the state, or 3) over the counter (OTC) for a short list of species, or 4) through an online race (sort of OTC) for "left over tags", 5) by getting called for a population management (depradation) hunt, or 6) win a some sort of fund-raising raffle or auction, or 7) win some other special raffle run by G&F.
The draw application deadline in 2013 was March 20 (I'm sorry, but I just found this forum). See page 15 (2013 edition) for list of licenses and fees. There are several ways to "game" the draw to improve your chances of drawing and there are resident/NR quotas attached to the draw.
Land-owner authorizations are available for elk, deer, and antelope (maybe some others). The state Dept of Game and Fish issues a list of land owners and authorizations can be found on Craigslist too. You pay the land owner to transfer the authorization, and then you pay the state for the hunting licence. Watch out for "ranch only" vs "unit wide" authorizatins; there are pros and cons both ways, and which depend on the ranch. No quotas here, money talks.
OTC licenses are available for turkey, bear & cougar subject areal and bag limits; oryx, barbary sheep, ibex outside the premium areas, and "fur-bearer" trapping.
Getting a licence through a raffle, and auction, or getting called for a depradation hunt is really a low-likelyhood event. Even buying a left-over tag on-line is very hard. There are almost never left over elk or deer tags.
Lemme know if this helps.