In New Mexico a municipality may call itself a: village, town, or city. (New Mexico Statutes §3-1-3) There is no distinction in the statutes and no correlation to any particular form (Mayor-Council, Commission-Manager, etc.).
Unless provided otherwise in a municipality's charter, municipal elections are held on the first Tuesday in March of every even-numbered year. (New Mexico Statutes §3-8-25) Elections are non-partisan (New Mexico Statutes §3-8-29C), and election materials (cards, signs, ads, etc.) are exempted from the requirements for all other elections that the responsible party be identified (as in "paid for by Committee to Elect Joe Candidate").
In addition to municipalities, limited local authority can be vested in landowners' associations and districts. An example of the former is the Madrid Landowners' Association, which is the closest thing to local government in Madrid, New Mexico. Its authority comes from the restrictive covenants that are written into all deeds.