Nuke on the Moon

In absence of atmosphere, the nuclear explosions will look very different.

The most notable effect will be the absence of fireball, and all subsequent effects relying on convective flows in the environment - no ascending mushroom. An above-Moon blast will look pretty much like an exoatmospheric blast, without the auroras. There will be a short intense pulse of gamma and light and heat, but no lingering thermal radiation as there will be no blackbody radiation of an incandescent fireball. Some induced radiation from the neutron shower will be on the surface. Neutron bombs may be still employed against structures without extensive shielding, as the lower damage and absence of blast wave will be actually advantageous here.

Surface and subsurface, that's a different song. Still strong differences, but actually something more will be visible. A significant amount of matter will be displaced. The trajectory of the ejecta, especially in the case of a subsurface blast, will be different than on Earth - no atmospheric braking, therefore virtually purely parabolic, and will travel much longer distance, due to lower gravity. Footage of dust plumes from the wheels of the moon buggy will provide some hints here.

For the subsurface blasts, the footages of the Sedan test may be useful, modified by the lack of atmosphere and lower gravity.
http://en.wikipedia.org/wiki/Sedan_(nuclear_test)

Surface blast may be a bit difficult to imagine. There will be a bright flash, virtually no fireball, some incandescent material scattered around, some surface area melted into glass. Much less amount of material displacement than for a subsurface blast.