OpenEXR is an open source image file format developed by ILM (VFX company owned my George Lucas responsible for the Star Wars films, the Pirates films, etc).
OpenEXR writes colour information as float values, which allow for finer precision when performing post work. Float is superior to Integer because of the amount of control you maintain over the image. For example, bmp has 256 levels (0, black - 255, white) in Red, Green and Blue colour space. Float while generally storing colour information between 0 (black) and 1 (white) can infact have negative values and values over 1. So a part of your image that may appear white, does have extra information in it.
Another way of looking at float values can be seen as follows. Say you have taken a photo of a white piece of paper in the sun, and the sun itself both in frame. When converting that image to a digital format for viewing on a monitor, the piece of paper and the sun, both will appear pure white, but we all know that the sun is brighter. A file format that uses integer will clamp the bright value of the paper and the sun to 255, but a file format that uses float may display the area of the image that has the paper at 1.0 (example), and the area of the sun as 1.8 (example). When brought into an image editing program capable of handling float values, decreasing the Gain or Gamma in the integer file format will just make the white turn grey for both the paper and the sun equally, while the float format will always keep the sun proportionally brighter than the piece of paper. Decreasing the gain by half would (in the example values given) make the piece of paper 0.5 and the sun 0.9.
Float precision can be either 32 bit (full float) or 16 bit (half float). OpenEXR also supports numerous extra channels like Alpha, Z-depth, Object ID, U, V, Normals, Velocity and more. It also supports a number of Compression formats including Piz (wavelet) which is growing in popularity.
If you are doing very little to no post work to your image, then saving it as a bmp file is sufficient. But if you are doing heavy post work, then it is wise to save it as an OpenEXR image, and use a compatible piece of software to process it.
When you render an image, TG2 (I assume) stores all pixel values as float internally, and when converted to a bmp file, will quantize the float value to the nearest integer. So a float value of 0.5 will become 128, 0.5039 will become 129 while any value float in between will either be rounded up or down to 128 or 129.
