Inside the mitochondria of animal or plant cells, a series of chemical reactions produce ATP. This ATP contains chemical energy, and is the main "fuel" that cells use to stay alive. Most of this ATP is produced by a series of proteins on the inner membrane of the mitochondria - these proteins collect electrons from NADH and FADH2 and pass the electrons down the protein chain ( hence "electron transport chain or ETC" ). This is shown in this digram from the plant physiology lab at the University of Lund, Sweden:

The membrane is shown in yellow.
The proteins are shown in green and orange/brown.
Notice that oxygen is used on the right hand side of the diagram: the oxygen receives the electrons at the end of the ETC. This is the reason you breath in oxygen: it keeps this whole system running, and provides the energy (ATP) that keep you alive.
The proteins actually pump hydrogen ions (H+ in the diagram) through the membrane. The hydrogen ions can only return through a special protein on the right of the diagram that produces ATP. The Nobel Prize for Chemistry in 1997 was won by the people who worked out the structure of this protein, and how it works. This diagram below shows this particular protein in more detail:

As you can see proteins are both complicated and beautiful.
There are a handful of facilities worldwide that can do this work, one is the Synchrotron Radiation Source at Daresbury, England.
For more details about this technique, click here.
The Daresbury site was used in this Nobel Prize winning work.
Coming back to the electron transport chain, each NADH molecule that goes through the chain produces 3 molecules of ATP, and each FADH2 molecule produces 2 molecules of ATP.
Altogether, for every glucose molecule that enters respiration, the ETC produces 32 ATP molecules, or 89% of the total ATP production.