From Reuters: UPDATE 3-Negative euro overnight rates show cracks in banking system
While there are many signs of cracks in the Eurozone, European banks “paying” negative rates to other banks for lending money is not one of those. It is simply the profit-maximizing consequence of the ECB’s decision to charge banks for any reserves above the required minimum.
What happens is the following:
Bank A (short of liquidity) must settle a payment with Bank B for one million euros. This means Bank A must send one million to Bank B. If Bank B believes they have enough liquidity already, they will carefully consider the consequence of being credited one million and being charged a negative annual 0,10% by the ECB.
A better alternative for Bank B is to ask Bank A not to send the amount due, and be paid a fraction of (up to) 0.10% for borrowing it.
See the Q&A on negative rates.