Real Exchange Rate vs PPP rate I'm having troubles to understand the difference between the Real Exchange Rate and the PPP rate.

I know the first one is calculated using a basket of goods and services so that the non-tradeable basket costs the same in 2 countries (imagining a 2-country world).  Now, isn't that the rate that would hold if PPP applies?

I think I'm wrong if I say that the real exchange rate is the one that makes PPP become valid.  Any help appreciated.
There are several exchange rate concepts that need to be distinguished. There are bilateral and multilateral (aka "effective") exchange rates, nominal and real exchange rates, and market-price versus PPP (purchasing power parity) exchange rates.


Let's start with the simplest concept: A bilateral nominal exchange rate, e.g., 117 yen/(US dollar), 0.8 pounds/euro, 7.7 (Swedish krona)/(Swiss Franc), etc. These are the exchange rates that are commonly reported in newspapers and in any number of online outlets. 


Aside: There are two distinct quoting conventions for exchange rates: the "E" system (with the home currency in the numerator) and the inverse, the "R" system. A nominal exchange of 1.2 Swiss francs per euro is expressed in the "R" system from the point of view of a euro area resident, and in the "E" system from the point of view of a resident of Switzerland. The "R" convention is conceptually easier to work with, since an increase of the numbers of foreign currency units per unit of home currency corresponds naturally to an appreciation of the home currency. Using the "E" system, the opposite is the case. There seems to be little or no standardization with regard to quoting conventions of bilateral exchange rates. However, multilateral exchange rates are always expressed using the "R" convention, such that a larger numerical value corresponds to an appreciation of the currency, and vice versa.

A bilateral real exchange rate is the bilateral nominal exchange rate multiplied by a ratio of price indices of the two currencies. (How the ratio is taken depends on whether the currency pair is quoted in the E or R method.) Naturally, many different price indices can be used; most common, though, is the use of headline CPIs (consumer price indices). Because CPI series are generally monthly (or even lower frequency) time series, real exchange rates too are generally monthly time series. The home currency appreciates in real terms against a foreign currency either if the home currency appreciates in nominal terms or if the home country's inflation rate is lower than that in the foreign currency.
While bilateral exchange rates are straightforward to report, interest often centers on multilateral exchange rates. The home economy has trade and financial ties not just with one other economy but (usually) with many different economies. To form an index of the multilateral exchange value of the home currency, it's necessary to weight the various bilateral exchange rates according to the importance of the foreign economies. Again, many different weighting schemes are possible; a common scheme uses merchandies trade weights based on imports and/or exports between the home economy and the most important foreign economies. Because bilateral exchange rates are ratios of relative prices, an exchange rate index (aka a multilateral or efffective exchange rate) is computed as a Tornqvist index (a geometric rather than arithmetic average).
Up to now, we've mainly dealt with market prices for both nominal and real exchange rates. While that's perfectly OK for many applications, it's not the most helpful way to proceed if one wants to make comparisons of standards of living across countries. Purchasing power parity (PPP) tries to facilitate such comparisons by figuring how much a standard "basket of goods" -- frequently, but not always, the goods in the CPI baskets -- costs in various countries. 

A famous case of a bilateral PPP exchange rate is the "Big Mac Index" of The Economist magazine: Since a Big Mac is pretty much the same in all countries where it's sold, one can work out the PPP-based bilateral exchange between the U.S. dollar (say) and all other countries as the exchange rates that would make the cost of purchasing a Big Mac the same. (Obviously, The Economist makes no claim that the Big Mac features prominently in anybody's consumption basket.) 

By comparing nominal and PPP-based exchange rates it is possible to make statements such as "currency A is overvalued (or undervalued) against currency B". Of course, different choices of consumption baskets will lead to different PPP exchange rates. Hence, depending on the choice of basket, conflicting answers may arise as to whether a currency is over- or undervalued.


For more about the Big Mac index, check out the Economist Magazine's own website.

For a non-technical discussion of the dollar indices used by the staff of the U.S. Federal Reserve Board see, e.g, the 2005 article Indexes of the Foreign Exchange Value of the Dollar in Federal Reserve Bulletin.