Everyone knows that spending from an address multiple times is bad, because Winternitz OTS.
But, what does that mean, exactly? What information is leaked in each signature, and how would an attacker exploit that information to discover the private key?
To expand on CFB's answer, because IOTA uses Lamport signatures, half of the private key is leaked each time. This halves the security level of the address (from 54 trytes of security to 27 trytes for a typical address with security level 2), making it exponentially easier to brute force the remainder of the private key with each key reuse. While after a single use funds are still relatively safe (27 trytes equates to about 128 bits of security), after 2 or 3 reuses, the funds will be in grave danger.
More information:
Lamport signature: How many signatures are needed to forge a signature?
Fragments of the private key of that reused address are leaked. An attacker could find a bundle hash (via brute-forcing) that can be signed with the leaked fragments. Luckily for the user, the window for the attack is short (only while the spending transaction is not confirmed). But if more iotas come to the reused address the attacker can spend them right away if he completes the brute-forcing by that time.
With every spend a random 50% of the private key is exposed. The reason that it is a one-time signature follows from what a second spend can reveal:
The overlap of the 2 random 50% reveals can be anywhere from 0% to 100%. Since it is random, the distribution for this will be on a bell curve.
Which makes 0% overlap (left/bottom of bell curve) and 100% (right/bottom of bell curve) overlap highly unlikely, 50% the most likely (middle/top of bell curve), and the likelihood decreasing faster and faster when you move towards either extreme.
But randomness being what it is, every value between 0% and 100% overlap can in practice occur. So if you are unlucky the overlap is a lot less than 50%, and in that case a lot more more than the theoretical average of 75% of your private key is exposed after the second spend.
This can be a problem because a possible attacker monitoring spent addresses can detect a second spend in progress, and until that second spend is confirmed he has time to brute force the missing parts of the private key and try to get his own spend of the funds in the address confirmed.
Any funds coming on that address after that are essentially gone the moment they arrive.
