A real return is the return on an investment, less the reduction in its value as a result of inflation.
real return = ((1+r) ÷(1 + i)) - 1 ≈r - i
where r is the nominal return over a period and
i is inflation over the period.
The nominal return is simply the percentage increase measured in currency. The approximation is accurate enough for valuation, except when inflation is very high.
Real rates are important as they tell you what the actual increase in value was, and how much of a return was just the effect of inflation.
Real interest rates are the most widely used type of real return.
Investors rarely think in terms of real returns. There is an obvious advantage in considering inflation for financial planning. There is another good reason to use real rates: for any kind of long term financial modelling real numbers are more stable than nominal numbers (see the Fisher hypothesis for an example). This means that they are often preferable for long term models such as those used for NPV valuations.