Under PMBOK methodology, how would you explain the difference between a 'project charter' and a 'project scope' to a stakeholder unfamiliar with the PM process?
Project scope is a conceptual thing describing the boundaries of your project, covering what the project encompasses and (maybe more importantly) what it does not encompass. From PMBOK 4:
Project Scope. The work that must be performed to deliver a product, service, or result with the specified features and functions.
Project charters are baselined documents used to capture the agreed-upon objective, scope, benefits, costs, milestones, governance, quality standards, etc etc etc of your project. From PMBOK 4:
Project Charter [Output/input]. A document issued by the project initiator or sponsor that formally authorizes the existence of a project, and provides the project manager with the authority to apply organizational resources to project activities.
I would explain that studies have shown two major clusters of reasons why projects fail.
Bad scope - people don't understand the boundaries of the work that needs to be done. Either they fail to do all the work needed to ensure that the project is ready for the customer, or they spend time and effort on things that are nice to have. Nailing down project scope is essential to ensure that the project involves the work, all the work and nothing but the work.
Bad sponsorship - projects succeed because the organization backs them, and the fail because the project team failed to ensure that they had all the support they needed. It does no good to develop a brilliant web site only to discover that the webserver cannot take any more traffic. There is a core team involved in the project, but they can be successful only if their work aligns with the organizations' interest and if the organization is interested in their success. That is the job of the project charter.
Generally the project charter incorporates a high level statement of the scope, but the scope is intended to set boundaries of work and the charter is intended to ensure that the organization backs the project.
If I could, I'd illustrate that with some examples of projects that failed because of scope and of sponsorship - examples that are relevant to the individual. (examples from my environment wouldn't be meaningful to anyone outside my environment).