What is Scrum?
Scrum is an iterative, incremental process for developing any product or managing any work. It produces a potentially shippable set of functionality at the end of every iteration. It's attributes are:

Scrum naturally focuses an entire organization on
building successful products. Without major changes -often within thirty days -
teams are building useful, demonstrable product functionality. Scrum can be
implemented at the beginning of a project or in the middle of a project or
product development effort that is in trouble.
Scrum is a set of interrelated practices and rules that optimize the development
environment, reduce organizational overhead, and closely synchronize market
requirements with iterative prototyes. Based in modern process control theory,
Scrum causes the best possible software to be constructed given the available
resources, acceptable quality, and required release dates. Useful product
functionality is delivered every thirty days as requirements, architecture, and
design emerge, even when using unstable technologies.
Over fifty organizations have successfully used Scrum in thousands of projects
to manage and control work, always with significant productivity improvements.
Scrum wraps an organization's existing engineering practices; they are improved
as necessary while product increments are delivered to the user or marketplace.
As heard about Scrum, "oh, that's just my idea X by another name". Except, Scrum
is spelled out as values, practices, and rules in a development framework that
can be quickly implemented and repeated.
Scrum has been used to produce financial products, Internet products, and
medical products by ADM. In every instance, the organization was logjammed,
unable to produce shippable products for such a long period of time that
engineers, management, and investors were deeply concerned. Scrum broke the
logjam and began incremental product delivery, often with the first shippable
product occuring with the same quarter.
Scrum as applied to product development was first referred to in "The New New
Product Development Game" (Harvard Business Review 86116:137-146, 1986) and
later elaborated in "The Knowledge Creating Company" both by Ikujiro Nonaka and
Hirotaka Takeuchi (Oxford University Press, 1995).
Based on their ideas and research in process theory performed in
collaboration with DuPont Advanced Research Facility,
Scrum was first formulated and presented to the Object Management Group in 1995.
The Scrum process is fully described in a recent book from Ken Schwaber and Mike
Beedle, Agile Software Development with Scrum (Prentice Hall, 2001).
An excerpt of that book can be read here.