the DX means it is those new line-up of lenses targeted at nikon's existing Dslr which have the 1.5x crop factor. basically they are made slightly differently to reduce side, weight, cost, etc and most importantly they are optimised as the main concerns is that they take the 1.5x crop factor into account.
on film body then also can be used but for a certain range of focal lengths (lower down the zoom spectrum) they cause vignetting on film body because film is fullframe (crop factor is 1x ie no crop at all) and the DX lens like as mentioned its made sligtly different than normal non DX nikon lens (dont have to dwell to exact into excatly how it is different than non-DX)