What is a vector vs. bitmap image?

I have found many of my clients through the process of working on a project come to ask this question: "What is a vector vs. bitmap image?" Many times I will ask them for vector versions of their logos and marketing materials to be re-purposed in the design of their web project. My question is many time met with puzzlement. I saw this video the other day of a master digital illustrator drawing over a bitmap image in Adobe Illustrator and thought this would be an excellent way to show the difference.. so for understandings sake:

The bitmap image is the photograph underneath which is confined by resolution and pixels. The vector image is the drawing being made over top in this video which is full scalable and editable to any size and essentially boils down to a mathematical equation. And because description still sounds so cryptic I offer this video for show and tell which is a time lapsed shot of many hours of work sped up to about 6 minutes:

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