"Contrail" is a shortened form of "condensation trail." An alternate name is vapor trail. One of the main byproducts found in the exhaust of any fuel-burning engine is water vapor. Water vapor is formed alongside carbon dioxide and carbon monoxide. This is true for jet airplanes, gas turbine engines and reciprocating engines. In the winter, you can see this water vapor behind the tailpipe or muffler as it condenses in the surrounding cold air.
In order for a contrail to persist, or to last for a lengthy time, it has to be formed above 26,000 feet with an outside air temperature of -40 degrees (Celsius or Fahrenheit; thermometers of either scale read the same at this temperature). Depending on the ambient conditions, a contrail may be short-lived, when you may see it dissipating a mile or two behind the airplane, or it may persist for hours, leaving the upper atmosphere crisscrossed by vapor trails.
Water vapor is one of the gases expelled from the exhaust of a jet engine. Provided that the surrounding air has a high enough relative humidity, the addition of further water vapor from the engine will cause the local relative humidity to rise above 100 percent, at which time the vapor condenses out and forms ice crystals. If the humidity is high enough, the contrails will persist for a lengthy time. When the relative humidity is very low, no contrails will form since the vapor contained in the exhaust gases is not enough to push the combined humidity up to 100 percent. When the humidity is in between, contrails will form but will quickly dissipate.
Some people use the character and persistence of contrails to forecast local weather conditions. A narrow contrail that rapidly dissipates signifies low humidity in the upper air, which can be taken as a sign of fair weather. Broad contrails that persist for a long time signify very humid air, and can be an early sign of a coming storm.