The test pyramid is a way of thinking about different kinds of automated tests should be used to create a balanced portfolio. Its essential point is that you should have many more low-level unit tests than high level end-to-end tests running through a GUI.
For much of my career test automation meant tests that drove an application through its user-interface. Such tools would often provide the facility to record an interaction with the application and then allow you to play back that interaction, checking that the application returned the same results. Such an approach works well initially. It's easy to record tests, and the tests can be recorded by people with no knowledge of programming.
But this kind of approach quickly runs into trouble, becoming an ice-cream cone. Testing through the UI like this is slow, increasing build times. Often it requires installed licences for the test automation software, which means it can only be done on particular machines. Usually these cannot easily be run in a "headless" mode, monitored by scripts to put in a proper deployment pipeline.
Most importantly such tests are very brittle. An enhancement to the system can easily end up breaking lots of such tests, which then have to be re-recorded. You can reduce this problem by abandoning record-playback tools, but that makes the tests harder to write.  Even with good practices on writing them, end-to-end tests are more prone to non-determinism problems, which can undermine trust in them. In short, tests that run end-to-end through the UI are: brittle, expensive to write, and time consuming to run. So the pyramid argues that you should do much more automated testing through unit tests than you should through traditional GUI based testing. 
The pyramid also argues for an intermediate layer of tests that act through a service layer of an application, what I refer to as SubcutaneousTests. These can provide many of the advantages of end-to-end tests but avoid many of the complexities of dealing with UI frameworks. In web applications this would correspond to testing through an API layer while the top UI part of the pyramid would correspond to tests using something like Selenium or Sahi.
I always argue that high-level tests are there as a second line of test defense. If you get a failure in a high level test, not just do you have a bug in your functional code, you also have a missing or incorrect unit test. Thus I advise that before fixing a bug exposed by a high level test, you should replicate the bug with a unit test. Then the unit test ensures the bug stays dead.
The Google Testing Blog expands on why you shouldn't rely on end-to-end tests.
Adrian Sutton explains LMAX's experience which shows that end-to-end tests can play a large and valuable role.
Kevin Dishman gave me the idea of adding the cost and speed arrows to the illustration
Most people know about the the Test Pyramid due to Mike Cohn, when he described it in his 2009 book Succeeding with Agile. In the book he refers to it as the “Test Automation Pyramid”, but in use it's generally referred to as just the “test pyramid”. He originally drew it in conversation with Lisa Crispin in 2003-4 and described it at a scrum gathering in 2004. Jason Huggins independently came up with the same idea around 2006.
7 Aug 2016: changed illustration
15 Nov 2017: added etymology
1: Record-playback tools are almost always a bad idea for any kind of automation, since they resist changeability and obstruct useful abstractions. They are only worth having as a tool to generate fragments of scripts which you can then edit as a proper programming language, in the manner of Twist or Emacs.
2: The pyramid is based on the assumption that broad-stack tests are expensive, slow, and brittle compared to more focused tests, such as unit tests. While this is usually true, there are exceptions. If my high level tests are fast, reliable, and cheap to modify - then lower-level tests aren't needed.