Normally, the APU is used on the ground to provide electrical power and air for heating, cooling, and engine starts.
The early jet airliners did not have APU's. The 727 is the first jet airliner that I know of that had an APU, but it was an afterthought on that airplane. So it is installed between the main landing gear wheel well and its shroud extends into each wheel well. It definitely had to be turned off before take-off because if you didn't, you lost all indication of whether it was running or not at liftoff and you first clue would be a wheel well fire warning alarm. The only way to turn it off in flight was to pull the fire handle for the APU, guessing of course that the APU was the source of your problem. Not a comfortable feeling.
There were attempts to add APU's to both 707's and DC-8's, but none of these worked very well.
On the twin jets of today, the APU is required equipment for ETOPS operations and it must be flight rated. It is used to provide a backup source of electrical power. Also, the APU must be operational with its generator on standby for Category III approaches. The idea being if you lost a engine driven generator while on the approach, the APU driven generator would come on line and not leave you in a lurch, low, slow, missing nav aids, in bad weather. So if your APU is inoperative on a twin, you cannot make a Category III approach.
Since the DC-9/BAC-111, the APU's have been flight rated on every airplane, twin engined or not, and it is almost always mounted in the rear of the fuselage. You can often see the APU exhaust extending from the airplane tail cone. On the trimotors, it may be located below the #2 engine, ala L-1011, DC-10, MD
At major airports, electrical power and airconditioning is now provided by a central unit that is more fuel efficient and cheaper to maintain than an APU and supplies; these services will handle several airplanes at a time.
At smaller airports, the APU is still used on the ground.