The term "West Coast Offense" actually came from Bill Parcels ,who coached defense led teams. After Parcels' Bills defeated the 49ers 17-3 in the 1985 playoffs he was quoted as saying "What do you think of that West Coast Offense now?"
The West Coast Offense s more of a philosophy of the game rather than actual set plays. Rather than setting up the running game first, this offense uses its short passes or draw defenders close to the line opening up long runs or passes later. The intent of this offense is to confuse the defense since most down and distance situations can be attacked by either a pass or a run. This forces defenders to be more passive since they cannot just focus on one specific play for a certain situation.
To use the West Coast Offense you need a quarterback who is extremely accurate since most of the throws will be in extremely tight coverage. You also need very sure handed receivers who aren't afraid to be hit. You also need a wide receiver who is blazing fast to be the deep passing threat. A quarterback such as Michael Vick have been very successful in this offense due to his speed and ability to run the football when need be.
The first NFL team to use this offense was the 1970 Cincinnati Bengals who were coached by Paul Brown with Bill Walsh at offensive coordinator. Currently 11 of the 32 NFL teams use this offense with varied success.