In Mexico you can either apply for a short-term or a long-term visa. The former is valid for up to 4 years, the latter does not expire.
There are two types of short-term residence visas (visa de residencia temporal) for stays of up to 4 years in Mexico. You can either apply for a temporary residence visa for students (residente temporal estudiante) or for a temporary residence visa (residencia temporal). Neither of them gives you the right to pursue remunerated activities in Mexico.
The long-term visa (visa de residencia permanente) is recommended (and even necessary) if you want to live in Mexico for more than four years. After getting a work permit, or if you have Mexican relatives or relatives with living in Mexico on a long-term residence visa, you can apply for a long-term residence visa yourself. This visa does not expire.
If you want to apply for a Mexican visa, you have to do so with the consulate or embassy in your home country. You have to be a national of the country where you apply. You will need the following: