"Contrails" is a contraction of "condensation trails" (as opposed to "chemtrails" - a contraction of "chemical trails" [think "trails" left by an agricultural crop duster aircraft as it makes an application pass over a field]).
All other things excluded, condensation trails can occur when specific conditions in the atmosphere occur - certain levels of moisture (visible or not), the mass of air that contains that moisture moving from one area of pressure to a lower area of pressure, and that mass of air moving at a velocity that is commensurate with producing a change-of-state of the moisture in that mass from a non-visible state to a visible state.
When non-visible-moisture-laden air moved rapidly from a high-pressure area to a low-pressure area a rapid change of air temperature takes place due to the expansion of the space between air molecules. In cool air water molecules clump closer together than they are in warm air, sometimes to the point that the moisture actually becomes visible.
All other things being equal, the faster the velocity of the moving air mass from the high-pressure area to the low-pressure area the more likely visible moisture will appear. That is one reason you can see visible moisture coming off the propellers of the C-130's in the photos which have been submitted to this website. The propeller tips are moving at nearly the speed of sound, their angle-of-attack relative to the air in front of them is such that they move a huge amount of air past them at a very high speed. The amount of moisture in the air is such that when it undergoes a compression/de-compression cycle as it passes the tip of the propeller blade is is quickly compressed then de-compressed and then is appears as vapor behind the propeller.
As one contributor has already explained "contrails" can also come from exhaust systems of piston-driven aircraft. Those "contrails" come from basically the same events - i.e., compression and then rapid de-compression of moisurized air. The most-often seen photographs/movies which show this effect are usually of B-29 aircraft. Most piston engines cannot operate at altitudes higher than about 20,000 feet because the air is too "thin" to sustain combustion in their cylinders. B-29's, like many other late-WW 2 aircraft - were equipped with "superchargers" - in effect very powerful "fans" which pulled in atmospheric air, compressed it, then sent it into the fuel/air intakes, thus providing sufficient oxygen to sustain the fuel burn in the cylinders. That "compressed" air naturally contained moisture - some of which did not completely get enough heat/pressure in the cylinders to be destroyed. It subsequently re-formed as condensation after it left the exhaust stack - only with a vengeance because of the extreme temperature shock it experienced going from the high temperature/pressure of the combustion cylinder to the sub-zero temperatures of altitude at the high altitude these bombers were flying at.