Spatial resolution in Remote Sensing is normally expressed in distance per sample rather than samples per unit distance. The same information is there, it's just expressed as the reciprocal of what is usual in normal graphics fields. The "per sample/pixel/whatever" is normally dropped as well and as it is considered implicit.
So a resolution of 0.5 m is really 0.5 m/px, which is 2 px/m or 0.0508 DPI.
This means that the pixels on the ground are squares that size (at least on average, under ideal conditions) A 0.5 m resolution raster data set has pixels that are 0.5 m squares of the Earth's surface. This gets more complicated once you start considering projection of the curved surface of the earth into a flat raster grid.