When driving in the Northern parts of Mexico, you see many cars with license plates from the United States. Some are people who cross the border into the United States frequently and some are permanent residents of Mexico and have brought their vehicle down from the U.S. While many people don't do this, Mexico actually has a process to legally bring a foreign plated vehicle into the country. (NOTE: If you do bring your car into Mexico, you must purchase a Mexican auto insurance policy as your insurance at home does not cover you.)
As a foreigner, you are allowed to import one U.S. or Canadian car into Mexico. If you are going to be in Mexico a short period of time (less than 180 days) you can get a temporary importation permit (TIP) as part of your tourist visa. The importation permit is tied to the visa, so keep in mind if you leave Mexico and leave the car, when you come back with a new temporary visa, the car will now be deemed in Mexico illegally, because the import permit is tied to the temporary visa, and it will not match. The only exception to this is if you have a Non-Inmigrante card (FM3), you can extend the import permit on your vehicle.
If you have become a permanent resident of Mexico, you cannot follow the process outlined above and it is illegal for you to bring in your car. You must go through an import broker to legally bring in the car. If you change from a temporary resident to a permanent resident, you actually have to export the car out of Mexico and then reimport the car using an import broker if you still want to keep the car in Mexico.
There are parts of Mexico where these rules vary. For instance, if you are in the Baja Peninsula, you are not required to have an import permit. You do need to keep your registration stickers current on the vehicle and keep in mind if you decide to drive from Baja into mainland Mexico, you will now be subject to the rules discussed above. This is also true for the state of Sonora.