These are three samples demonstrating dither. The set below is the above set with levels adjusted to highlight the effects of dithering.
From left to right, first one is most visibly banded; no dither present. The second one is the same gradient with dither; much better. And the third one, the one on the right, has dither, and I have additionally added a layer of noise, in effect boosting the dither. See commentary for why you would add the additional noise. See "additional notes" for how to dither transparency/alpha channel.
I have been making large renderings of gradients with minimal color variation, and banding was very visible without dithering, like in the first sample. Then I found out about dither and it improved things a lot, that's the second sample. But on a large rendering (less visible here) there was still some banding. I needed more aggressive dither. So I copied* the gradient layer, rasterized it and added a tiny amount (0.1%) of uniform monochrome noise. That improved the large scale image considerably. That's the third sample. The improvement from additional noise (which is what dither is essentially) is less visible on a small scale like in this sample here, but it was very important on a large rendering like the ones I was working on.
*I created a separate layer for noise because that way I could control noise with the new layer's opacity.
I've only discussed the actual image. The hardware and driver side of things can be influential as well; another device might display things differently. Other answers have covered that pretty well.
Photoshop does not dither the alpha channel. If you need dithering on transparency you have to dither the alpha channel manually. I would use a mask layer to create transparency, so it would only be a matter of dithering that mask layer. And when working with layer masks don't forget to ALT+left click on the layer mask to display it.