Negative interest rates are a sign of the financial system not working normally. Usually, if you want to borrow money from someone, you pay them more money for the privilege. When interest rates are negative, you pay someone for the privilege of lending them money. It sounds insane and goes against common sense, but it happens - and is happening at the moment, as our financial system has not been working normally for over five years - for a few reasons.
There are several examples of negative interest rates in Europe at the moment, where the eurozone debt crisis has fantastically distorted financial markets thanks to the "will they, won't they" political shenanigans of the European Union. The basic problem that countries like Greece and Spain (and if you're reading this in, oh I don't know, 2013, then France) have is that no-one wants to lend them money. All of the big banks that used to lend money to Greece and Spain for a profit are now too scared to do so, so they look to lend it elsewhere where they can still make a profit without the risk of losing all their money in a default.
All of this money sloshing around looking for a safe haven has led to record-low borrowing costs for Germany and Britain, and has even led to a mad dash to sell euros in exchange for currencies that look safer, such as the Danish crown and Swiss franc. If the eurozone implodes, then the last guy holding euros is going to look like a right idiot, and no-one wants to be that guy: so plenty of people are just getting out of euros altogether.
You might think this was great for the countries who are trusted by the markets, and in one sense it is. They can borrow money easily and all those people buying their currency means their economic prospects are generally viewed as healthy - Denmark, for instance, benefits from close links to the German powerhouse. But there are downsides to everyone buying your currency, too. Currency values are set by supply and demand, and if everyone buys the Danish crown then the crown becomes very expensive, making it much harder for Danes to export their goods. An even more fundamental risk is that all of those Danish crowns end up getting invested into the Danish economy, fuelling an uncontrollable property bubble in a repeat of the process that happened to the western economy as a whole before the credit crunch.
Taking these risks into account, Denmark recently made a decision - if all of these scared investors want the privilege of buying the Danish currency to keep their money safe, then by heck they can pay for it. They started charging banks for the privilege of lending crowns to the Danish central bank, which is a common way to park money to keep it safe. This decreases the attractiveness of crowns to investors, and might even lead them to invest their money elsewhere in a way that might help the European economy as a whole. It's an interesting way of forcing banks to lend to businesses rather than hoarding their money - although, in this case, the primary motivation is to stop European investors buying so many Danish crowns that they become too expensive. Switzerland did something similar in the 1970s when the oil crisis made their currency an attractive safe haven.
The other example of negative interest rates in Europe in recent years has been the German government's borrowing costs. Germany is the strongest economic power in Europe, and at points, investors have been so worried about a complete meltdown of the eurozone that they were paying the German government for the privilege of lending them money. This wasn't a calculated policy tool as in the Danish case, as government borrowing costs are set by the market as a whole. But it was a sign of just how scared European investors were of lending to anyone else that they were willing to lose some of their money just to have somewhere safe to park the rest of it.
These are two specific examples, but negative interest rates could have broader uses - for instance, when an economy is in what's called a liquidity trap, which means that the economy is so broken that standard tools of monetary policy fail. This happened during Japan's lost decade and may well be happening to Europe now. Negative interest rates set by a central bank stimulate spending because they penalize the hoarding of money and encourage everyone to spend it - if your money actually loses value while in the bank, you're going to want to spend it fast. We're a way away from such compulsion yet. But don't be surprised to hear someone suggest it - or even implement it - when we really start going to hell in a handbasket.