Cameras with more dynamic range allow you to take photos with dramatic differences in highlight and shadow areas while retaining detail in both.
Think of a brilliant sunset on a rocky beach: Bright sunset in the background, with dark rocks in the foreground. High dynamic range means more of the extremes will be faithfully reproduced.
Dynamic range test data courtesy of DxO Mark.
A35 test data on DxO Mark GX-20 test data on DxO Mark