Nikon D610 has a 24.0MP Full frame (35.9 x 24 mm ) sized CMOS sensor and features Expeed 3 processor. On the other hand, Nikon D80 has a 10.0MP APS-C (23.6 x 15.8 mm ) sized CCD sensor . Nikon D610's sensor provides 14MP more than Nikon D80's sensor, which gives a significant advantage in real life. You can print your images larger or crop more freely.
On the other hand, please keep in mind that Max sensor resolution is not the only determinant of resolving power. Factors such as the optical elements, low pass filter, pixel size and sensor technology also affects the final resolution of the captured image.
Below you can see the D610 and D80 sensor size comparison.
As seen above, Nikon D610 has a 2.3x Larger sensor area than Nikon D80. 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.