When it comes to the oxidation with NAD⁺, remember that the Nicotinamide group of NAD⁺ is the site of reduction. We're going to say that a basic residue, denoted as b, of an enzyme abstracts an H⁺ from the hydroxyl group of a substrate. What happens here is that the hydride transfers to NAD⁺. If we take a look, we have our residue, basic residue. It would abstract this hydrogen here.

Oxygen would hold on to the electrons to make a double bond here, and at the same time, this hydride ion would leave and attach itself to this carbon of NAD⁺, causing a movement of this pi bond here and a movement of this pi bond to the nitrogen. As a consequence of this, our substrate is now oxidized, and we have a carbonyl here, and then NAD⁺ is reduced to NADH. So here goes the hydrogen, the hydride ion that attached, the pi bond moved here, and then the other pi bond became a lone pair on the nitrogen here. This is how we look at when it comes to NAD⁺ oxidizing our substrate and as a consequence being reduced to NADH.