The Zero Bound Problem is a monetary policy limitation that a central bank might encounter during a recession. It occurs when the central bank lowers interest rates to 0% and would optimally lower them further, but cannot do so because it does not have the power to set a negative interest rate (if interest rates were negative, everyone would choose to hold their money as cash instead of investing it). The zero bound is troubling because it restricts the actions that a central bank can take to combat a recession. The problem may be lessened by maintaining moderate inflation, because if the inflation rate is positive a central bank can attain a negative effective (real) interest rate. As an illustration, suppose that inflation is 3%, and the central bank sets the (nominal) interest rate to 0%. A saver in this economy invests $100, and receives $100 back at the end of the year. However, due to inflation this is only worth $97 at the end of the year, showing that the effective (real) interest rate was -3%.
The zero bound is not just a theoretical possibility, but has actually arisen several times in recent years. Japan encountered the zero bound in the midst of a severe recession in the 1990s and 2000s and the US Federal Reserve now faces the problem during the Great Recession. Keynesians take the zero bound problem seriously because they believe a central bank should have as many tools as possible available to respond to a recession.