What is "ISO" in general, and how is the scale defined?
How does the ISO scale for film speed differ from ISO sensitivity as used in digital cameras?
Is lower ISO always better?
Answer
In photography, ISO generally refers to a measure of "Film Speed", which I use including reference to digital sensor sensitivity.
In short, the actual letters ISO are a name for the International Organization for Standardization (not, officially, an acronym -- more information here), and in photography it refers to the ISO 12232:2006 standard and other related standards: ISO 12232:1998, ISO 5800:1987, ISO 6:1993 and ISO 2240:2003. (Links on the Film Speed page.) Film historically has also used the ASA and DIN standards, the former using the same numbering system, and DIN using an entirely different scale.
The standards for film and digital are technically different (in ways I haven't investigated closely enough to report fully on), but generally they're similar enough that for practical purposes, they're essentially the same (notwithstanding Reciprocity Failure, which many films are quite prone to, though digital generally is not). So if you measure an exposure with your digital camera, you could use that exposure with a film of the same rating as the ISO setting you used in the digital camera, and expect to get a similar exposure (unless the shutter speed is long or short enough for Reciprocity Failure to kick in for the film) (also assuming similar equipment otherwise -- no differences in filters, etc.).
In both digital and film, a higher number indicates greater sensitivity. A number twice as high is twice as sensitive (e.g. 200 is twice as sensitive as 100, 400 twice as sensitive as 200, etc.). So, when shooting in relatively low light, and wanting a relatively-fast (e.g. fast enough to stop motion) shutter speed, a higher sensitivity rating will be essential (so no, lower is not always better!)
In digital cameras (and similarly but differently in film), higher ISO ratings tend to create noise (the related effect in film is increased graininess). So, while it's not always better (that depends on what you're going for), lower ISO ratings are always (or at least almost always) lower in noise, which may be desirable. (In the case of low light photography where shutter speed is not a concern, long exposures mixed with lower ISO ratings will create a "better" image -- though it's conceivable that some may like the effects of the noise; certainly there's appeal at times to film grain.)
As for how the scale is defined, it is based on measurements of an image produced based on a certain scenario of illumination. The details are complex, so I'll leave the details as an exercise for the reader. A lay summary is that (for digital) it's a measure of how quickly the digital sensor becomes "saturated" with light. (For film, the process is related but different.)
In summary: higher ISO is more sensitive but noisier (but not necessarily worse), digital and film rated for the same ISO (or ASA for the film) will have similar sensitivity, and the scale is based on how quickly an image will become "saturated" given a certain amount of illumination.
NOTE: I'm gearing up to do some experimentation related to the controversial answer from Matt Grum. Hopefully, my results will create a nice clear noise-free answer to the important point he brings up: that a high-ISO image with a low amount of light will be less noisy than a low-ISO image with the same amount of light getting to the sensor, which is later amplified in post-processing. More to come, hopefully in... EDIT: Well, I've failed to make this happen for a while now. I may still do it at some point. In the mean time, I'll also point to this article that talks about comparisons of native versus non-native ISO values, and amounts of noise in them, which, while the article doesn't exactly say so, I think is probably directly related to exactly this question.
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