Spoiler warning for those who haven't beaten Fallout 3
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Here is how the ending system in Fallout 3 basically works. The ending that you receive is mostly based on these three aspects: Your Karma level (Good, bad, neutral), what you did at the end of the game (sacrificing yourself, sending Lyons in to sacrifice herself, or sitting around and doing nothing while time runs out), and what you decided to do during your confrontation with President Eden at Raven Rock during the quest "The American Dream" (siding with President Eden, refusing his proposal of eliminating all mutation from the wasteland, and/or implementing the modified F.E.V that he gives you.)

Completing sidequests might provide you with some extra images to show the effect that you've had on the wasteland, but they won't affect the actual dialog of the ending.

To answer your question of how many endings there are though, it's been said that there are about 12 unique endings. However, they're not as unique as you might think they would be. A single sentence from the narrator or a single image might be the only difference between two endings. Check out the Mahalo page on the Fallout endings to view some videos and for more information.
Source(s):
http://www.mahalo.com/Fallout_3_Ending
and my experience through playing the game

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