The electron transport chain is an important process in the making of ATP (adenosine triphosphate) which is the energy used by the cells of an organism.
The production of ATP occurs in the mitochrondria, organelles located inside the cell.
The mitochondrian has 2 membranes, an outer one, and an inner one with many invaginations.
The space inside the inner membrane is called the matrix.
It is along the inner membrane that the electron transport chain works.
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NADH and FADH2 are reductants, molecules that carry electrons.
One of them starts by giving 2 electrons  to a protein and becoming NAD+ (or FAD).
The proteins are able to pick up and pass on the electrons to other proteins (such proteins are called cytochromes).
As the electron pair moves down the electron transport chain, from protein to protein, they attract 3 pairs of protons 
which move across the membrane into the intermembrane space (2 pairs for FADH2).
After the last protein, the electron pair joins with an oxygen atom and 2 other protons to form H2O, or water.
After a while, as more and more protons are stored in the intracellular (or intermembrane) space, an H+ gradient is created due to the greater amount of protons on one side than the other.
As a result, potential energy is stored until the protons move back into the matrix through the protein on the right, ATP synthase.
As the protons pass, the ATP synthase uses their potential energy to create ATP.
One ATP is created for each proton pair that passes. So, NADH results in 3 ATP and FADH2 results in 2 ATP.
As well, one water molecule is produced each time.
Thus, the electron transport chain is an integral part of cellular respiration because it is the major site of ATP formation.