FM3 - Non Immigrant 
When you do NOT want to seek permanent residence in Mexico.
For vacations and casual trips to Mexico, simply fill out and use the tourist permit (FMT), available from the airline you travel with or the port of entry* (See below)
For Work Placements: If you plan to live and work in Mexico, then a Visitor's Permit (FM3 Visa), renewable annually, is probably your best option.
For Other Activities: You should apply for a FM3 permit commensurate with your activity - e.g. Student, Journalist, Professional, et al.
*To holders of passports from specific countries only.
If you are in doubt, contact your nearest Mexican Consulate.
FM2 - Immigrant,
Economically Active 
When you want to acquire permanent residency AND you want to work in Mexico:
You will need to satisfy the requirements for entry (e.g. professional, sponsored by a company, etc -- see notes above about FM2), OR be able and prepared to invest at least 26,000* times the minimum daily wage in Mexico City and apply for an FM2 visa.
FM2 - Immigrant,
Not Economically Active
When you want to acquire permanent residency but DO NOT want to work in Mexico:
If you are of retirement age (50+), and have at least US$1,000 or equivalent income per month (plus an additional US$500 per month for each dependant, e.g. spouse), then a Retiree FM2 visa will be the most straightforward route. NB: There is no 'official' minimum income, by law, that you need to prove; however the sums mentioned above are a rule-of-thumb and commensurate with the levels of income most retirees are being asked to prove.
If you are not of a retirement age (below 50) and want to live but not work in Mexico, you will need to contact the Mexican Consulate. Provided that you can prove a steady income, you may be granted a FM2 visa to live in Mexico and seek permanent residency. You will need to state what you intend to do there, e.g. early retirement due to health, etc.
*Sums are subject to change. Contact your local Mexican Consulate for the latest policy on investor's visas.
FMT - Tourist Permit
Coming soon ...