A: Great questions!
First we need to distinguish between nominal and real interest rates. The article What's the difference between nominal and real? The quick answer is that nominal interest rates are the ones you typically hear about (prime rate, etc.) whereas real interest rates factor out inflation.
This week we will examine zero nominal interest rates. Next week we will look at zero real interest rates.
Zero Nominal Interest Rates
A zero nominal interest rate occurs when the interest rate is the same as the inflation rate. If inflation is 4% then interest rates are 4%. If you lent or borrowed for a year at a zero real interest rate, you would be exactly back where you started at the end of the year. I loan $100 to someone, I get back $104, but now what cost $100 before costs $104 now, so I'm no better off.Typically nominal interest rates are positive, so people have some incentive to lend money. During a recession, however, central banks tend to lower nominal interest rates in order to spur investment in machinery, land, factories, etc. If they cut interest rates too quickly, they can start to approach the level of inflation. Inflation will often rise when interest rates are cut, since these cuts have a stimulative effect on the economy.
According to some economists a zero nominal interest rate can be caused by a liquidity trap:

"The Liquidity trap is a Keynesian idea. When expected returns from investments in securities or real plant and equipment are low, investment falls, a recession begins, and cash holdings in banks rise. People and businesses then continue to hold cash because they expect spending and investment to be low. This is a selffulfilling trap."
As you might expect this is more of a theoretic possibility than something than occurs regularly in the real world. However, it did take place in Switzerland in the late 1970s, where investors bought negative nominal interest rate bonds because of the strength of the Swiss franc.
I hope this answers your question about negative real interest rates. In What Happens if Nominal Interest Rates Go To Zero? we examine the case of negative nominal interest rates. If you have a question about interest rates, please contact us, by using the feedback form.