They helped me to think much clearer about the different concepts, disciplines and fields of software testing.
The original concept is from Brian Marick, but the quadrants are discussed in depth in the brilliant book "Agile Testing" by Lisa Crispin and Janet Gregory.
There is also a presentation from Agile Vancouver 2008 held by Janet Gregory.
A project can draw different benefits from each of the testing quadrants:
Traditional software testing focuses almost exclusively on the right side, critiquing the product but not playing a productive part in supporting the creation of the product.
Traditional software testing is involved late in the development process to detect bugs but not to prevent them.
The lower left quadrant is the typical developer testing. Unit-testing and integration-testing, hopefully automated and with continuous integration, are quite common practice today.
The upper left quadrant is where we are entering quite new territory compared to what is common practice in the industry.
It's here where we place Acceptance Test Driven Development (ATDD) and Behavior Driven Development (BDD). Those are quite new agile methodologies that focus on improving quality by placing activities from testing early in the development process. Those activities are constantly supporting the development process throughout its whole duration.
However it's important to understand that all testing quadrants are important. It's not that focusing completely on one quadrant can make the other quadrants completely obsolete. However a well tailored testing strategy uses all quadrants appropriately to improve testing as a whole. Depending on the concrete project the different quadrants have to be wighted accordingly.