Updated 2/22/18, link here: Residence Renewal / Starting the Process
So start off with much thanks to Ramona on MistCutter who laid the ground work for all this.
* n.b. this was all done before the USA policies on deportation were instituted so we'll have to see how Mexico responds and if our process would've been significantly different next year
Normal visits to Mexico by USA citizens include a Tourist Visa good for a stay up to 180 days, if you've been to Mexico via plane you've seen it. The form you fill out and give to customs at the Mexico check-in has a piece they tear off and give to you. When you leave they take this back. This is your tourist visa. The problem for us was that this would require a trip out of the country every 180 days or less, which means 4 airplane tickets per year. Ouch.
The 'Residente Temporal' or Temporary Resident card, allows you to stay in country indefinitely as long as you renew. After 11 months but not more than 12 months you can renew for 1, 2 or 3 years (some choose the 3 year option because after 4 years you can apply for permanent residency)
From here I was going to write all about our experience but I found this website and it does a great job of explaining things (sorry the formatting changes in the blog). Where applicable I have added my comments: