What process produces both NADH and FADH2?

In both prokaryotes (like bacteria) and eukaryotes (like humans), sugars are broken down (catabolized) to release energy that can be stored in the bonds of ATP, the energy currency of the cell.  

This process starts with glycolysis which occurs in the cytoplasm of a cell.  During glycolysis a 6-carbon glucose is broken down into 2 3-carbon pyruvate molecules.  The process also nets 2 ATP (4 produced, but 2 used during the chain) and 2 NADH molecules.

The next step involves the oxidation of both of the pyruvate molecules to produce 2 acetyl CoA molecules.  These molecules are able to enter the TCA cycle (also known as the citric acid cycle or Kreb's cycle).  Acetyl CoA combines with oxaloacetate to form citric acid to begin the cycle.  During this cycle, the cell produces 4 CO2, 6 NADH, 2 FADH2, and 2 ATP per glucose molecule.  The process takes place in the mitochondria of eukaryotes and cytoplasm of prokaryotes (they do not contain mitochondria).  

The electron carriers NADH and FADH2 then take their electrons to the electron transport chain.  This is in the inner mitochondrial membrane of eurkaryotes and the cell membrane of prokaryotes.  The electron carriers are oxidized as they deliver their electrons to electron carriers in the chain.  Electrons are passed from carrier to carrier of increasing electronegativity and eventually to water (for those organisms using aerobic metabolism).  Along the way, some electron carriers pump protons across the membrane creating a proton gradient on one side of the membrane.  The only part of the membrane through which they can diffuse down their concentration gradient and back across the membrane is through the enzyme ATP synthase.  As they pass through the protein, the energy captured is used to produce ATP.  The total number of ATP produced by all steps is about 38 in bacteria and about 36 in eukaryotes.