The Greek catapult was developed by Dionysius the Elder of Syracuse around 350 BC. Dionysius actually called the machine Katapaltai, and they are mentioned in the Siegecraft (Poliorkētika) treatise of Aeneas Tacticus. Originally, because of their large bulk and problems with portability, they were used mainly in seige warfare. Alexander the Great is thought to be the first to use them for cover fire for ground troops in addition to seige warfare, essentially introducing artillary to the military world.
While Rome is also known to have use artillary (the ballista and onegar mainly), their standard use in battle dates from the time of Julius Ceasar.
Catapults soon went out of use with the introduction of the trebuchet in Medieval times, and went out permanently with the widespread use of gunpowder shortly thereafter. At this time, cannons became the weapon of choice in seige situations.
The catapults themselves were generally constructed of wood and were assembled on site by engineers. There was usually not much in the way of materials transported, as most time wood could be chopped down from nearby trees for use in construction.