How long / to what temperature should I let meat rest?
The resting period is not generally done by an absolute temperature&mdash;you would have measured for (near) your target temperature before before taking the piece out of the oven or off the griddle or whatever.

Traditionally, it should be proportional to the size of the cut or roast.


5-10 minutes is good for a steak or individual chicken pieces, depending on their thickness
Medium size roasts and smaller poultry, say 10-15 minutes
Large roasts and large poultry (like a turkey), up to 30 minutes.


If you do want to measure by temperature for more precise results, Kenji Alt of Serious Eats recommends based on practical experimentation, for steaks:


  Ideally, no matter how well-done you've cooked your meat, you want to
  allow it to cool down until the very center has reached 120°F (49°C).
  At this stage, the muscle fibers have relaxed enough that you should
  have no problem with losing juices.


He also reviews some of the science, and various explanations for why the resting works.

Cook's Illustrated's Americas Test Kitchen performed an experiment with several roasted pork loins, cutting them with no resting, or resting of 10, 20, 30, and 40 minutes.  They found that even a 10 minute rest cut down on juices expressed by 60% compared to the unrested roast.  They found far less difference in the roasts that were rested for longer periods, although there was continued improvement in the juice retention.  They summarize (original emphasis):


  So what to do? Curb your enthusiasm and let your meat rest. But not
  for too long. The most dramatic decrease in moisture loss for these
  pork loins took place during the first 10 minutes of rest. Additional
  time helps—but not if that means your dinner will be cold. If you’re
  cooking a big roast, you can wait longer—about 30 minutes. Thin steaks
  should only wait 5 to 10 minutes before you dig in.


This shows that the traditional advise on resting times is actually fairly effective.

Note:  Thanks to Jefromi for sharing this reference.