Basically, a shift right means the hemogoblin is going to drop off oxygen at a faster rate than the normal curve. This happens when you have a higher CO2 level, H+ level, more BPG (allosteric inhibiter for hemoglobin+oxygen).
A shift left, means that the hemogoblin has a higher affinity for oxygen, meaning that it will take up more oxygen than normal. Infants generally have this shift compared to their mother, as O2 will more readily diffuse into their blood cells. High altitude animals like llamas have this also.
In your example an elephant with somehow a permanently shifted curve left would be able to thrive at lower O2 levels compared to a "normal elephant". It's the opposite for the mouse.
For practical purposes, in a normal system, the shift to the right (temporarily) just lets you dump off more oxygen from hemoglobin to muscle tissues that are overworked (producing more H+)