What happens inside the mitochondria?

You can think of mitochondria as a eukaryotic cell's "power plants". They generate a supply of "usable energy" in the form of ATP. Mitochondria are important actors in aerobic respiration, which is a long process that ultimately results in the production of ATP.

Here are the basic steps of aerobic respiration [1]:

1. Food (ingested) + Air (inhaled)
   
2. Carbohydrate + Oxygen and Nitrogen
3. Glucose + Oxygen (final products of digestion and inhalation)
4. ATP (energy) + Carbon Dioxide (exhaled) + Water (exhaled and excreted)

Mitochondria take the stage at step 4 (aka cellular respiration), as they produce adenosine triphosphate (ATP). For simplicity, I won't be talking about anaerobic respiration, and will be stick to eukaryotic animal cells.

ATP is the cell's energy carrier or energy store molecule:

See the three phosphorous groups connected by oxygens to each other? That's the triphosphate section of the molecule. Each of these oxygens has a negative charge, and thus repel each other.

These bunched up negative charges want to escape - to get away from each other, so there is a lot of potential energy here. If you remove just one of these phosphate groups from the end, so that there are just two phosphate groups, the molecule is much happier [2].

...which we can visualize as the following process:

This conversion from ATP to ADP is an extremely crucial reaction for the supplying of energy for life processes. Just the cutting of one bond with the accompanying rearrangement is sufficient to liberate about 7.3 kilocalories per mole = 30.6 kJ/mol. This is about the same as the energy in a single peanut.

Living things can use ATP like a battery. The ATP can power needed reactions by losing one of its phosphorous groups to form ADP, but you can use food energy in the mitochondria to convert the ADP back to ATP so that the energy is again available to do needed work [2].

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Now that we know our ADP from ATP, let's familiarize ourselves with the basic structure of a mitochondrion.

Click the image to enlarge.

Source: Wikimedia

The mitochondrial matrix (which "lies" upon the inner membrane) is lined with proteins involved in electron transport and ATP synthesis, such as the enzyme ATP synthase [3].

So how do we get from glucose to ATP?

1. Glycolysis: After food is digested, we have a fresh supply of glucose. Glucose metabolism in all cells produces two molecules of pyruvate from each molecule of glucose. This occurs outside the mitochondria, usually in the cytoplasm.
2. Cellular respiration: Occurs in different sub-compartments of the mitochondrion. Each pyruvate molecule is converted to three molecules of carbon dioxide, and the energy produced by this reaction is "trapped" in ATP (i.e. the energy is used to fuel the production of ATP).

There are three sub-pathways of cellular respiration:

1. Pyruvate oxidation.
2. Krebs cycle (also known as citric acid cycle, or tricarboxylic acid cycle) - occurs in the matrix of the mitochondria
3. Electron transport chain

If you're really curious about steps 1 and 2, watch this Khan Academy video for a beginner's guide to the Krebs cycle:

I'll discuss step 3, i.e. the electron transport chain [3]:

A) The "freed" high energy electrons produced by the Krebs cycle travel from one protein complex to the next in the mitochondria's inner membrane. This process is known as the electron transport chain. At the end of this electron transport chain, the final electron acceptor is oxygen, and this ultimately forms water (H20).

B) Simultaneously, the electron transport chain produces ATP. As the chain passes the electrons along, the energy released causes hydrogen ions (or protons) to be pumped out of the mitochondria's matrix space. This creates an gradient which now drives hydrogen back in through the membrane again, via the ATP synthase. As this happens, the enzymatic activity of the ATP synthase causes the creation of ATP to ADP.

Whew. Here's a diagram of the electron transport chain, to summarize points A and B. Remember, we're still inside the mitochondrion (click the image to enlarge):

© 2010 Nature Education All rights reserved [3].

And that's a bare-bones primer to the magical world inside our mitochondria!

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I've kept this guide as user-friendly as possible, since that's what the question seems to be asking. As such, I didn't talk about anaerobic respiration in eukaryotic cells.

Let me know if you'd like more information, if you think I could simplify something, or if you'd like me to be more precise. I appreciate the feedback - thanks!