Interest rates typically run a couple of percentage points higher than the inflation rate. (I'm talking about prime rates here; what you pay your credit card company is between you and Mastercard). Having interest rates higher than inflation means that if you borrow money and wait to pay it back, it costs you something and, conversely, that if you lend somebody money, you will realize a profit when they pay you back with interest. This accords with common sense and many economic models presume that the world must always be this way.
After all, in a world with negative interest rates, a bank would charge you money every month for leaving funds in their care, and would pay you to borrow money from them. It is difficult to imagine such a bizarro world lasting for long.
However, the governments of most developed countries nowadays rely on monetary policy almost exclusively in controlling economic growth and inflation. (Fiscal policy comes into it somewhat, but has lost favor for a variety of reasons not worth going into here). This means that when growth is low, the governments cut interest rates to spur borrowing and drive new investments in factories, etc. Since cutting interest rates tends to increase the amount of money in circulation, this policy typically increases inflation. This is not usually a problem, as inflation is typically low during a recession. However, sometimes recessions happen along with high inflation, and the results are ugly.
In a prolonged recession, the government may cut interest rates over and over again. If they cut rates down to zero (or close to zero) and inflation remains, then real interest rates become negative.
Two examples of negative interest rates are worth discussing. During the 1970s, Switzerland offered negative interest rates, but only to foreigners. This weirdness emerged because of significant speculative interest in owning Swiss francs, on which the foreign investors expected to make so much money that they were willing to pay for the privilege of holding the currency. Economists note the complexity of this example and typically treat it as a special case that does not break existing models.
The second example is Japan, which has had negative interest rates in the international markets intermittently since 1998. For most of this time, these were merely negative nominal rates, possible because the Japanese economy has been in deflation through this time as well. If you are borrowing at -1% and your salary contracts at -2%, the real cost of borrowing is still a positive 1%.
However, on January 24, 2003, normal Japanese domestic interest rates went negative. Banks lent money on Friday night and accepted back a smaller amount on Monday morning. Why would they do this? Presumably to reduce their Yen exposure at a time when confidence in the Japanese economy, and the Japanese banking system in particular, are very low. A total of about 15 billion yen ($127 million) was traded in that weekend at a negative rate.
Given prolonged recession in the U.S. and many European economies in 2001 and 2002, and the standard monetary response of declining rates in all of them, the possibility of Japan-like developments elsewhere must be considered. Economists are accordingly spending more time on the issue of negative interest rates than in many years. See http://www.ledr.com/neg_rates/neg_rates_home.htm for some examples.
Sources:
http://www.economicsuk.com/archive/industry/2001-dec.html
Japan Economics Institute Weekly Report, 13 November 1998
Reuters News Service wire report January 24, 2003, 2:32AM ET