Taking advantage of one-time pad key reuse? Suppose Alice wants to send encryptions (under a one-time pad) of $m_1$ and $m_2$ to Bob over a public channel.  Alice and Bob have a shared key $k$; however, both messages are the same length as the key $k$.  Since Alice is extraordinary lazy (and doesn't know about stream ciphers), she decides to just reuse the key.

Alice sends ciphertexts $c_1 = m_1 \oplus k$ and $c_2 = m_2 \oplus k$ to Bob through a public channel.  Unfortunately, Eve intercepts both of these ciphertexts and calculates $c_1 \oplus c_2 = m_1 \oplus m_2$.


  What can Eve do with $m_1 \oplus m_2$?  


Intuitively, it makes sense that Alice and Bob would not want $m_1 \oplus m_2$ to fall into Eve's hands, but how exactly should Eve continue with her attack?
There is a great graphical representation (which I found on cryptosmith, but they keep changing their url structures, so I've added the graphics in here) of the possible problems that arise from reusing a one-time pad. 

Let's say you have the image



and you encrypt it by using the binary one-time-pad (xor-ing on black and white)

.

You get the following extremely secure encryption

.

If you then encrypt a smiley face with the same one-time-pad,



you get another secure encryption

.

But if you have both and you xor them together



then you get the image



which, as you can qualitatively and intuitively see is very insecure.

Reusing the same key multiple times is called giving the encryption 'depth' - and it is intuitive that the more depth given, the more likely it is that information about the plaintext is contained within the encrypted text.

The process of 'peeling away' layered texts has been studied, as ir01 mentions, and those methods improve with more layers.