What does the real exchange rate represent?
The real exchange rate represents a comparison of the purchasing power of two countries' currencies. This is different from the nominal exchange rate, which simply tells how much of one country's currency you can get by exchanging a unit of another country's currency.
With a nominal exchange rate, you know how many Euros, for example, you can get for one US dollar. But that does not really tell you all you need to know. What should be of interest is how much you can actually buy with that dollar as compared to the Euros you could get for it. Real exchange rates measure this.
The real exchange rate takes that dollar and measures how much you can buy with it in the US. It then takes the number of Euros that you can get for a dollar and sees how much you could buy with that in the EU. The comparison is the real exchange rate.
So, you might have a case where the dollar is nominally weaker than the Euro (where $1 gets you less than one Euro) but where you can actually buy more in the US with the $1 than you could buy in the EU with the fraction of a Euro. In that case, you would know that the dollar really has more value that the nominal exchange rate says it does.