There are two concepts here.

First is the size of the image circle. In Nikon's line-up :
- a "DX" lens has a smaller image circle and can be used ONLY with an APS-C size sensor (or with the D3/D700 which knows about cropping the image)
- a "FX" (or unspecified) lens has a larger image circle and can be used with cameras with an APS-C sensor and a sensor the size of a piece of 35mm film

In practice a "DX" lens is only used with digital cameras because there are no film cameras with APS-C sized films. An "FX" lens could be used with film or digital

Second the idea that a lens is optimised for digital rather than film. There are a number design parameters that could be affected:
1. High resolution. The light-sensitive elements on a sensor are much smaller than the grains in a film grain. It takes four elements to make one pixel and they are spread out across the sensor not piled up as in a film
2. Low chromatic aberation. This is the purple fringing you get on the edge of bright lines that destroys the impression of sharpness in the image
3. Light hits the sensor head-on rather than at an angle. Film can accept light at any angle but on a sensor the corners of the image tend to go dark if the light is arriving at too steep an angle (this is less of a problem nowadays)
4. Rear lens element is treated to avoid reflections. The sensor is shiny. You can get a ghost image in extreme conditions from light coming out of the lens, being reflected off the sensor then back off the rear lens element onto the sensor again