OpenEXR is a fairly new format developed for High Dynamic Range imagery (HDRi) by Industrial Light and Magic (ILM). It was created to streamline visual effects work in a number of ways.

First, it can use several different kinds of compression, both lossy and lossless, for efficiency with a variety of different kinds of data. One particularly useful compression algorithm is called PIZ, which is very good at compressing grainy images.

Second, it permits several different bit depths, including 32-bit integer and 32-bit floating point.

Third, with high bit-depths comes the ability to store and work with linear light information. Most digital images have a gamma correction applied that expands the color space where the human eye is most sensitive and reduces color space in ranges where it is less sensitive. While this makes the images look better when displayed, it really futzes with the math during image processing. A discussion of gamma and linear workflows is a bit beyond the scope of this post, so I'll leave it at that. For more information on that topic, I recommend this article: http://mymentalray.com/wiki/index.php/Gamma

Fourth, it has a theoretically unlimited number of available channels, each of which is (ideally) tagged with its purpose. In addition to an alpha for transparency, an OpenEXR file can also contain channels serving a variety of different purposes. In visual effects, this is commonly used to store multiple render passes of 3d objects in a single file, so a compositor only needs one image sequence instead of a dozen (or more). Also, certain utility channels can be generated, such as a z-depth channel that can be used to create depth of field or atmospheric perspective tricks during post-processing.

Fifth, it is an open source format available under a free license, which encourages developers to create tools to make use of it.

OpenEXR has its downsides, though. Pulling an arbitrary number of channels out of a single file tends to be slower than using multiple files, particularly when some of those channels may not even ultimately be used. It is also not well supported in all applications yet. Software that sees a lot of use in the film industry uses it, but other applications have been slower to adopt it. Most users will never have the need to use the high bit-depth available from OpenEXR, and even fewer will ever need more than four channels. So there is a lot of added complexity with very little benefit. That may change over time, as people come up with more reasons to use those extra channels, but I expect it will be many years before end users are swapping files with a .exr extension.