The $NADH$ can be compared to a basket filled with a hydrogen ion and two electrons. It can transmit these electrons so act as reducer . The $NAD^+$ is comparable to the empty basket. It can recapture both electrons and hydrogen ion (dehydrogenate!), so act as oxidant.
1 : adenine 2 : ribose 3 : nicotinamide 4 : diphosphate group Hence the name: NicotinamideAdenineDinucleotide
The exchange $H^+$ and 2 $e^-$ happens in the pyrimidine nucleus of nicotinamide.