What type of engine/motor is used in aircraft APU?
Since the only fuel available on an airliner is kerosene, the APU is a kerosene-powered jet. Using a different and maybe more efficient type of engine would require an independent fuel system, which does not make sense for the small savings possible.

From the 737.org page on APUs:


  There are many different APUs available for the 737. The most common
  is the Garrett GTCP (Gas Turbine Compressor [air] Power unit
  [electrics]) 85-129. This was standard for the series 1/200 but when
  the -300 was introduced it was found that two to three times the
  energy was needed to start the larger CFM56 engines. Garrett produced
  the 85-129[E] which had a stretched compressor, ie the impellers were
  lengthened and the tip diameters increased. When the 737-400 was
  introduced, even more output was required and Garrett produced the
  85-129[H]. This has an Electronic Temperature Control which limits hot
  section temperatures depending upon demand and ambient temperatures.
  By 1989 the 85-129[H] was the most standard APU in all 737 models,
  although there are actually 14 different models of the 85-129 in
  service with 737s.


To answer your question about the 737 NG, it continues:


  Other APUs available are the Garrett GTCP 36-280(B) and the
  Sundstrand APS 2000 on the 3/4/500; and the Allied Signal GTCP 131-9B
  for the NGs. The main difference between them is that the Garrett is
  hydro-mechanical whereas Sundstrand and Allied Signal are FADEC
  controlled.


Be sure to load the 737.org page, it has lots of pictures and data on the APUs.