Why is it that when you look in the mirror left and right directions appear flipped, but not the up and down?  
Here's a video of physicist Richard Feynman discussing this question.

Imagine a blue dot and a red dot.  They are in front of you, and the blue dot is on the right.  Behind them is a mirror, and you can see their image in the mirror.  The image of the blue dot is still on the right in the mirror.

What's different is that in the mirror, there's also a reflection of you.  From that reflection's point of view, the blue dot is on the left.

What the mirror really does is flip the order of things in the direction perpendicular to its surface.  Going on a line from behind you to in front of you, the order in real space is 


Your back
Your front
Dots
Mirror


The order in the image space is


Mirror
Dots
Your front
Your back


Although left and right are not reversed, the blue dot, which in reality is lined up with your right eye, is lined up with your left eye in the image.  

The key is that you are roughly left/right symmetric.  The eye the blue dot is lined up with is still your right eye, even in the image.  Imagine instead that Two-Face was looking in the mirror. (This is a fictional character whose left and right side of his face look different.  His image on Wikipedia looks like this:)



If two-face looked in the mirror, he would instantly see that it was not himself looking back!  If he had an identical twin and looked right at the identical twin, the "normal" sides of their face would be opposite each other.  Two-face's good side is the right.  When he looked at his twin, the twin's good side would be to the original two-face's left.

Instead, the mirror Two-face's good side is also to the right.  Here is an illustration:





So two-face would not be confused by the dots.  If the blue dot is lined up with Two-Face's good side, it is still lined up with his good side in the mirror.  Here it is with the dots:



Two-face would recognize that left and right haven't been flipped so much as forward and backward, creating a different version of himself that cannot be rotated around to fit on top the original.
