Electron Transport Chain in Mitochondria
The good ol'electron transport chain (ETC).

Before beginning let us begin by looking at the structure of a mitochondrion and the purpose of the ETC.


(source: tokresource.org)  

Take of note four things: 


The intermembrane space
The matrix
The electron transport chain proteins and
ATP Synthase


Now lets talk about the purpose of the ETC, essentially its job is to create a H+ (proton) gradient between the intermembrane space and the matrix it does this by taking protons from the matrix and "pumping" them out to the intermembrane space.

This creates a higher concertation in the intermembrane space and lower concertation in the matrix. This gradient will result in a process called chemiosmosis where these H+ ions (protons) will go through ATP synthase on the inner plasma membrane in order to return to the lower concentration in the matrix.



So the most important thing to note about this is that the flow of protons through the ATP Synthase is what gives it the energy necessary to create ATP.

Alright now that is out of the way lets discuss how the ETC goes about pumping these electrons out of the matrix into the intermembrane as well as how ATP Synthase works.

If you were to zoom into the area that is labeled electron transport chain proteins (that I told you to note) you would see something like this.  



What we see here is that ETC is actually made up of a series of complexes (I, II, III, and IV) and a couple of "electron/reaction" carriers (ubiquinone and cyt c). These complexes and carriers are in charge of moving an electron via a series of redox reaction along resulting in the "pumping" of protons out the matrix at each complex.

*Note, understanding redox reactions help in understanding what is happening here.

The chemistry gets a little intense at this point, so I'm going to quote and link the Wikipedia article on this, which does a wondrous job explaining what occurs complex to complex.


  In Complex I, two electrons are removed from NADH [the electron carries from the previous steps] and transferred to a
  lipid-soluble carrier, ubiquinone (Q). The reduced product, ubiquinol
  (QH2), freely diffuses within the membrane, and Complex I translocates
  (pumps) four protons (H+) across the membrane, thus producing a proton
  gradient.


Note, that in Complex II no protons are pumped into the intermembrane.


  In Complex II additional electrons are delivered into the quinone pool (Q) originating from FAD [Again, electron carriers from previous steps].


Complex III receives the reduced ubiquinones from Complex I and II. 


  in Complex III, two electrons are removed from QH2 at the QO site and sequentially transferred to two molecules of cytochrome c, a water-soluble electron carrier located within the intermembrane space. The two other electrons sequentially pass across the protein to the Qi site where the quinone part of ubiquinone is reduced to quinol. A proton gradient is formed by one quinol (2H+2e-) oxidations at the Qo site to form one quinol (2H+2e-) at the Qi site. (in total four protons are translocated: two protons reduce quinone to quinol and two protons are released from two ubiquinol molecules).


Finally 


  In Complex IV, four electrons are removed from four molecules of cytochrome c and transferred to molecular oxygen (O2), producing two molecules of water. At the same time, eight protons are removed from the mitochondrial matrix (although only four are translocated across the membrane), contributing to the proton gradient.


NADH+H+ → Complex I → Q → Complex III → cytochrome c → Complex IV  → O2

Article here

So after all this beautiful chemistry we are left with a proton gradient. This is when ATP Synthase comes into play. Once the proton "flows" through it will bind ADP and inorganic phosphate together to form ATP.

energy + adp + pi → atp



More on how the ATP Synthase works here

This is water down in the actual chemistry portion of it but I feel like it will give you a better grasp on how it all works. If you have anyways especially regarding the series of reactions between complexes feel free to ask!