Self-Testing Code is the name I used in Refactoring to refer to the practice of writing comprehensive automated tests in conjunction with the functional software. When done well this allows you to invoke a single command that executes the tests - and you are confident that these tests will illuminate any bugs hiding in your code.
I first ran into the thought at an OOPSLA conference listening to "Beddara" Dave Thomas say that every object should be able to test itself. I suddenly had the vision of typing a command and having my whole software system do a self-test, much in the way that you used to see hardware memory tests when booting. Soon I was exploring this approach in my own projects and being very happy with the benefits. A couple of years later I did some work with Kent Beck and discovered he did the same thing, but in a much more sophisticated way than I did. This was shortly before Kent (and Erich Gamma) produced JUnit - a tool that became the underpinning of much of thinking and practice of self-testing code (and its sister: TestDrivenDevelopment).
You have self-testing code when you can run a series of automated tests against the code base and be confident that, should the tests pass, your code is free of any substantial defects. One way I think of it is that as well as building your software system, you simultaneously build a bug detector that's able to detect any faults inside the system. Should anyone in the team accidentally introduce a bug, the detector goes off. By running the test suite frequently, at least several times a day, you're able to detect such bugs soon after they are introduced, so you can just look in the recent changes, which makes it much easier to find them. No programming episode is complete without working code and the tests to keep it working. Our attitude is to assume that any non-trivial code without tests is broken.
Self-testing code is a key part of Continuous Integration, indeed I say that you aren't really doing continuous integration unless you have self-testing code. As a pillar of Continuous Integration, it is also a necessary part of Continuous Delivery.
One obvious benefit of self-testing code is that it can drastically reduce the number of bugs that get into production software. At the heart of this is building up a testing culture that where developers are naturally thinking about writing code and tests together.
But the biggest benefit isn't about merely avoiding production bugs, it's about the confidence that you get to make changes to the system. Old codebases are often terrifying places, where developers fear to change working code. Even fixing a bug can be dangerous, because you can create more bugs than you fix. In such circumstances not just is it horribly slow to add more features, you also end up afraid to refactor the system, thus increasing TechnicalDebt, and getting into a steadily worsening spiral where every change makes people more fearful of more change.
With self-testing code, it's a different picture. Here people are confident that fixing small problems to clean the code can be done safely, because should you make a mistake (or rather "when I make a mistake") the bug detector will go off and you can quickly recover and continue. With that safety net, you can spend time keeping the code in good shape, and end up in a virtuous spiral where you get steadily faster at adding new features.
These kinds of benefits are often talked about with respect to TestDrivenDevelopment (TDD), but it's useful to separate the concepts of TDD and self-testing code. I think of TDD as a particular practice whose benefits include producing self-testing code. It's a great way to do it, and TDD is a technique I'm a big fan of. But you can also produce self-testing code by writing tests after writing code - although you can't consider your work to be done until you have the tests (and they pass). The important point of self-testing code is that you have the tests, not how you got to them.
Increasingly these days we're seeing another dimension to self-testing, with more emphasis put on monitoring in production. Continuous Delivery allows you to quickly deploy new versions of software into production. In this situation teams put more effort into spotting bugs once in production and rapidly fixing them by either deploying a new fixed version or rolling back to the last-known-good version.