What is meant by real interest rates?

Real interest rates are interest rates - inflation

It is important to calculate real interest rates because if inflation is higher than the interest rates, then real interest rates may be negative (as described in the extract). When real interest rates are negative, people's money is losing value (even when in banks) which can lead to higher spending, as people want to spend their money before it de-values, which can be disastrous as it may lead to even higher inflation rates.