I'll try to avoid getting too technical - there are others that can fill in more detail. I think, apart from resolution, the most important aspects to me would be dynamic range and low light performance. DxO rates sensors on the following:

Resolution

With sharp lenses and good technique, more pixels can never hurt, especially if you are printing large or cropping wildlife images for example. Apart from those two scenarios, higher resolution is something you usually don't see unless you are pixel peeping, and comparing two similar spec cameras may be hard to distinguish the difference.

Colour Depth

Sensors with more colour depth can resolve more tonal values. I would think that the colour spaces most people use would limit the ability to see real life differences. DxO rates colour depth under the "Portrait" score.

Dynamic Range

I have shot a few frames with a D800, and it's incredible what detail you can pull from both the shadows and highlights compared to older cameras. This is easily seen in real life high-contrast images such as landscapes. In a few years we can ditch HDR, bracketing and throw out our grad ND filters :) DxO rates dynamic range under their "Landscape" score, as extended range is useful to avoid blown out skies.

Low Light (High ISO) performance

This is another measurement that's easy to see in real life examples, as some sensors are certainly better at limiting noise in low light. DxO rates low light performance under their "Sports" score, given its usefulness in poorly lit gyms, stadiums and hocky rinks.