The Fuji X10 camera has two auto modes. The EXR, I would call a more advanced auto setting then the usual auto mode of typical point and shoot cameras. It is even more complex then the standard auto mode that exists in this camera. This mode takes advantage of the CMOS sensor in the camera & lets the sensor change the configuration of individual light-gathering points on the sensor.
EXR is proprietary technology and a specific term used by Fuji to describe how it's sensor works. It provides for even greater automatic decision making capabilities within the camera. Now that might be thoroughly confusing, and one might ask why have two auto settings, but if you go test out the Fuji Xl10, you would see what has promoted the interest in thread.
Beyond these humble comments, I am pretty ignorant about the what, why, and how of all of this. I guess we need to ask Fuji. It just seems to provide quality results with minimal effort in a very small complex camera.
There are many discussions of this feature on the internet, however here is one on Flickr that might be of interest: