35mm Sensors
First let’s get the well-known 35mm format out of the way. The exact dimension of a “35mm sensor” measures 36×24mm, which doesn’t seem to have any logical relation. That’s because the name is in reference to 35mm film, which was the most common film size before digital cameras came around. It was named because it was 35mm in width, however the sprockets and information that needed to be included on the film resulted in an image size that was roughly 36×24mm. Obviously, digital sensors don’t need the sprockets and frame information, but the name stuck so that DSLR manufacturers could market their cameras as having the same size sensor as their traditional film cameras. (Thus you could use the same lenses and expect the same field-of-view.)
Inch Fractions
Other digital camera sensors get their size names and numbers from a much more bizarre history. When video cameras were invented back in the 1950s, they used vacuum tubes to capture the image, much like televisions of the day used vacuum tubes to display the image. These sensors were measured by the outside diameter of the tube, which typically produced a usable image about 2/3 of that size.
When smaller digital sensors were developed, they decided to measure them based on what the tube size would be if it were a 1950s video camera. As a result, camera sensors that measure themselves this way (1", 1/3", 1/2.8", etc.) typically have a diagonal measurement roughly two thirds of that diameter. However, to make matters even worse, manufacturers often wanted the sensor size to be a simpler fraction, so instead of being a more exact 9/16" or 27/32" for example, measurements were often rounded up or down to the nearest simpler fraction. This means it’s possible to have multiple digital sensors that are all classified as 1/3" sensors but with different actual dimensions.
As more smaller digital sensors needed more differentiation, decimals were added for the measurements between 1/3" and 1/2", and between 1/2" and 1". We now have sensors categorized into various measurements involving fractions with decimals (1/1.9", 1/2.8", etc.), but all are still rough approximations of what the diagonal tube size might be. To make matters even worse, the practice of measuring sensors based on their diagonal measurement means that sensors with a widescreen aspect ratio fall into a different size category than a sensor that's the same width but in a 4:3 aspect ratio. Meaning a smaller widescreen sensor can offer a wider field of view than a larger “fullscreen” sensor.