There are two basic kinds of Corp runs. Acquisition runs are when the Johnson asks you to recover a package that was taken from one of his couriers. Extraction runs are when the Johnson mentions a client wanting a career change but being held at his current company, and it is your job to find the client within the corp building and escort him back out the main entrance. The different levels of difficulty correspond to the corp that you are sent into -- see each entry for building-specific tips.