Dona Ana is an old village just north of Las Cruces. Though there is some renovation and restoration going on there, it's still a bit down-at-the-heels. A significant part of the old town is taken up with a pretty ratty trailer park, though some of the oldest structures are being restored. A work in progress is a good way to put it. I wouldnt choose to live there....just a personal choice.
Mesilla is a village with it's own town government, police and fire/rescue and it's own history. The boundaries of the old town are, roughly, Avenida de Mesilla, Calle Sur, Calle Oeste, and Calle Norte. The town limits have expanded a lot recently to include more open farmland, including many pecan orchards, and residential land all the way up into the West Mesa, just across the Rio Grande.
Las Cruces and Mesilla are adjacent. Las Cruces is the only "city" of any significant size in Dona Ana County, though there are smaller towns all the way down the Mesilla (Rio Grande) Valley to El Paso.
This website may give you a bit more details than the Mesilla town government website:
www.oldmesilla.org