How does jet engine water injection work?
On many early jet airliners water was sprayed it into the engine air intake or compressor where evaporation cooled the air and allowed cooler, denser air to be ingested by the engine. The result was increased thrust, especially needed for operation from hot and high airfields, or a cooler temperature in the hot section of the engine at the same thrust. From SAE Paper Number 2004-01-3108:


  Water injection was first used over 45 years ago on commercial
  transport aircraft to increase thrust. Boeing 707-120 aircraft with
  Pratt &amp; Whitney JT3C-6 engines and later, the Boeing 747-100 &amp; 200
  aircraft with Pratt &amp; Whitney JT9D-3AW and -7AW engines, all used
  water injection for thrust augmentation.


This report discusses various techniques for water injection:


ahead of the low pressure compressor. This is the common method for thrust increase in hot and high conditions.
ahead of the high pressure turbine. This is used in cold weather, when injection ahead of the low pressure compressor would run the risk of ice accretion.
together with fuel into the combustor. This technique is commonly used with stationary gas turbines in order to reduce nitrogen oxide emissions.


And NASA-TM-2010-213179 is more specific on the system:


  Four water-injected Pratt &amp; Whitney JT3C–6 engines were used on early
  Boeing 707–120 Stratoliner aircraft to augment takeoff thrust on days
  above 20 °F. This system used a belly tank to store demineralized
  water and an electrically driven boost pump to deliver water to the
  four engines. At that point, an engine-driven mechanical pump
  increased the pressure to about 400 psi for injection before the
  low-pressure compressor (LPC) on days as cold as or colder than 40 °F
  and for injection into the high-pressure compressor (HPC) only on days
  between 40 and 20 °F.


What would happen if kerosene instead of water was filled into the water tanks was tried out by a BAC-111 on September 6, 1971:


  The aircraft collided with a bridge, shearing off both wings, after a
  double engine failure occurred during takeoff. The water-injection
  system to cool the engines during takeoff was inadvertently filled
  with kerosene instead of water.


More recent work on water injection focused on the reduction of nitrogen oxides which form in lean combustion and high temperature. By lowering the temperature level of the combustion process, the nitrogen oxide pollution can be greatly reduced.