Both Nikon P7800 and Fujifilm X20 have 12.0 MP resolution sensors but Nikon P7800's sensor is 1/1.7" (7.44 x 5.58 mm ) and Fujifilm X20's sensor is 2/3" (8.8 x 6.6 mm ).
Since Fujifilm X20's has a larger sensor area with the same resolution, this means that it also has a larger pixel area hence better light collecting capacity for a given aperture compared to Nikon P7800.
Below you can see the P7800 and X20 sensor size comparison.
As seen above, Fujifilm X20 has a 1.4x Larger sensor area than Nikon P7800. Larger sensors give photographer more control on the depth of field and blurry background compared to smaller sensor when shot in same focal length and aperture.