When I use the term 'agile testing' I'm talking about a set of good practices which has helped me and my teams deliver higher quality software, whether or not we were part of an agile team. We know that people, not tools or methodologies, are what makes projects succeed. Anything we can do to allow ourselves to do our best work fits my definition of 'agile'.
'Agile testing' practices come about by applying agile values and principles to testing. For example, communication is essential, and by collaborating with our customers up front to write test cases, we can improve communication. By driving coding with customer-facing tests, we're more likely to deliver what the customer wanted.
Some other examples: Automating our regression tests allows us time to do exploratory testing, so we strive for 100% regression test automation (I've never achieved anything close to this on a team who isn't automating unit tests, but we can still automate some testing). Exploratory testing is inherently agile so we work at getting better at that. Collaborating helps communication, so testers should get together with programmers and with customers from the start of a project, as well as pairing with each other.
Simple design is important in designing and automating tests. Refactoring applies to any type of coding including test scripts. Retrospectives are the single most important practice I can think of for any team, because you can't improve without identifying and focusing on what you need to improve. Dividing work into small chunks, having a stable build at all times (because we have continuous integration and collective code ownership), releasing business value frequently, all that allows us to do a better job of testing.
None of this is really new, and a lot of it I was doing years before I heard of XP, Scrum, agile, etc.
I haven't come up with a good elevator speech for agile testing so I have to resort to examples. I've heard people come up with some great one-liners but then I can't remember them. I think the Agile Manifesto works from a testing point of view as well as from a coding point of view.
Testers assist the developers and the rest of the project community
in identifying what is desired, what we have, and the differences between
them, with the four points of the Agile Manifesto as guiding principles.