To elucidate on silverlake's writeup a bit, the critical path is the "spine" of any given project. It takes its name from Critical Path Management - a complex school of "scientific" project management developed during the second world war on the Manhattan Project and the methodological foundation for all contemporary project management.
In some ways, it's nothing more than a list of the order in which job stages must be completed, but since large scale projects often have a great deal of work happening in parallel it's more a listing of bottlenecks in production. As the various side sub-projects flow together, the "big river" that they follow to the sea is the critical path.
The best way to graphically examine the critical path is with a PERT chart, or "Program Evaluation and Review Technique". Most any project management software package, like the creatively named Microsoft Project will automatically create a PERT chart from other project info, like a Gannt chart (which is essentially a fancy outline and time estimate). In a PERT chart, the individual job stages appear as boxes (nodes), with their interrelationship illustrated by lines (vertices). Using connectivity and adjacency logic, the software picks out the critical path and highlights it. Add up how long the it takes to complete the critical path, and you have the shortest possible time in which to complete the project, as you have collapsed out the processes that can be pursued in parallel.
I must say that if you has told me in high school that I would wind up a project manager at an advertising agency, I might have punched you out. However, the word "critical path" is a real show stopper in a management meeting. Throw out a gem like, "The rest of this is just window dressing! We have got to get back on the critical path!" Then one should authoritatively whack the PERT chart. The PERT chart already put the fear of god into them. Once you invoke the wrath of the Critical Path, everybody shuts up.
Frederick W. Taylor