The PCM in our cars operates in two modes: Open Loop and Closed Loop. Ignition timing is adjusted in response to engine rpm, manifold absolute pressure, engine coolant temperature, throttle position, transmission gear, etc. During open loop operation (while motor is cold), the PCM receives input signals and responds according to preset PCM programming rather than using O2 sensor readings. Open Loop is important because O2 sensors do not work properly until they've warmed up a bit from the exhaust gases (and hence the purpose of the "air pump"). During closed loop operation, the PCM monitors O2 sensor signals and adjusts the injector pulse and ignition timing based on the readings from the warmed-up O2 sensors. The PCM then attempts to maintain the ideal AFR by fine tuning fuel injection. This produces the best performance (fuel economy, lowest emissions, etc) for a given engine load.

Also, contrary to some opinions, it is true that the vehicle will not drop out of closed loop with a 160 degree thermostat since that depends primarily on O2 sensor temperature, not engine temperature (http://dodgeram.org/tech/gas/Trouble/O2_sensor.htm, http://www.fierolt1.com/faq.htm).

Shoebox, correct me if I'm wrong!

 

Heh, I'm not here to grade anybody.

I would say that the AIR pump is to reduce emissions and warm up the cat(s), not the O2's (though when the cats warm, so do the sensors).

Closed loop can be affected by several factors, but purely by temperature, it is normally reached by ~140º.