Class D = an amplifier that switches the output devices really fast between positive and negative rail. The analog signal, a bit like DSD in that sense, is obtained using an LC low pass filter. Very efficient in power.
Class G = non switching amplifier (usually class A or AB) with several (more than 1) different power supply rails. The sole purpose of this technique is increasing efficiency, heat. Carver PA amps worked that way decades ago already. Small rack sized amps with huge power and little cooling needed.
Class H is well.. kind-off a class G variant. The difference with class G is that while class G has more than 1 power rails (so in 'steps') the class-H amp has a (switching) power supply that creates an internal voltage rail (actually 2, 1 positive and 1 negative) that does not switch (like class G) nor is constant (A, AB) but is modulated (when voltages above a certain voltage) are needed and thus always is just a few volt higher than the required output voltage.
The advantage is even higher efficiency than class-G. Almost on par with class-D.
Normal class AB (or even class A) amps can be created with these techniques while wasting of heat is reduced (higher efficiency)
class G and H are not related to class-D but the rail voltage modulation is switching so class-D-ish.