ISO is one of the three legs of the exposure triangle used to make sense of what goes into determining an exposure. The other two legs are aperture and shutter speed. ISO is the sensitivity to light as pertains to either film or a digital sensor. A lower ISO number means less sensitivity and a higher ISO number means more sensitivity. (With a caveat regarding digital cameras, see below.) Film has a single ISO rating, meaning that if you put a roll of ISO 400 film in a camera, you will be shooting at ISO 400 for the entire roll. Digital sensors can be set to various ISO speeds depending on camera model. For example, a Canon 5D Mark IV’s native (not including expanded ISO) ISO range is ISO 100 - 32,000. One stop of light, in terms of ISO, is equal to either double, or half, the current number. For example, IS0 100 is 1 stop darker than ISO 200, while ISO 400 is one stop brighter than ISO 200.
Explanation of ISO
The acronym ISO itself is a reference to the International Organization for Standardization. However this organization does far more than define camera sensitivities, it promotes universal standards for measurements of all different types, on an international level.
Fun fact: Instead of calling themselves the IOS, the title "ISO" is in reference to isos, (ίσος) which means "equal".
Previously, film sensitivity was also measured a similar way by another organization, the ASA or American Standards Association. This has been superseded by ISO in modern times, but the measurement itself and the scale remain effectively the same.
How Is ISO Measured?
They way ISO is measured, by the International Organization for Standardization itself, (ISO 12232:2006) is simply a specific level of brightness or exposure.
This brightness level is, visually, 18% grey. Does this mean that ISO 100, 200, 400, and others all corresponds to fixed brightness levels, such as lumens or EV? Unfortunately the answer is NO, in the real world. In every manner that ISO is referenced on charts and graphs, it is simply used as a corresponding brightness based on your shutter speed and aperture. (The Exposure Triangle)
For example: ISO 100, 1 second, and f/1.0 correspond to the fixed brightness "EV 0". However, that same brightness level could also be achieved at ISO 200, 1/2 second, and f/1.0, or ISO 400, 1/4 second, and f/1.0. Or you could change both your aperture and your shutter speed at the same time, and use any ISO setting you desire, yet still be able to arrive at the same final EV brightness of 18% grey.
Low ISOs and High ISOs
Many modern digital cameras start counting their ISO at ISO 100, and go up from there. Since one stop or EV worth of ISO is either double or half, from ISO 100 the common ISO's would be as follows: 100, 200, 400, 800, 1600, 3200, 6400, 12800, 25600, 51200, 102400, and so on and so forth.
Fun fact: Officially, the International Organization for Standardization has not actually defined any ISO's higher than 10,000 yet, as of 2013. Digital cameras, on the other hand, have "natively" gone many EVs higher than this, thanks to cleaner and cleaner gain amplification.
Are higher digital ISOs actually more sensitive?
One common mistake that photographers make is how they describe the way ISO works on digital cameras. For example, when the ISO is raised from 100 to 200, that sensor is not actually becoming more sensitive to light. In other words, it is not actually collecting more photons than it was before. The sensor is collecting photons the same way, however it is amplifying the signal that these photons are creating on the sensor. This does not mean that shooting at higher ISOs is pointless, of course, for numerous reasons. For more information, see the High ISO Definition.
Actual ISO versus stated ISO
In theory, on all camera sensors, as well as all films, ISO settings should be exactly the same. Each camera that uses ISO 100, for example, should receive exactly the same brightness of exposure. (Assuming identical shutter speeds and apertures.)
Unfortunately, not all camera sensors achieve such precision, and some digital cameras are actually slightly more sensitive or less sensitive to light. In other words, ISO 400 may actually behave like ISO 300 on a certain camera, for example. Generally speaking these discrepancies are no larger than 1/3 or 1/5 of a stop or EV, however.