What are some real-world examples of the different types of trading blocs? The subsequent types of trading blocs have the following definitions:

Free Trade Area -- free trade among members
Customs Union -- free trade among members and common external tariff
Common Market -- free trade among members, common external tariff, and free movement of factors of production
Economic Union -- free trade among members, common external tariff, free movement of factors of production, and harmonization of all economic policies

What are some real-world examples of these different types of trading blocs?
There are actually five different types of trading blocs: preferential trading areas, free trade areas, customs unions, common markets and economic (and monetary) unions [1].  

A preferential trade area (PTA) is a trading bloc that reduces tariffs on particular goods between two countries but does not abolish them completely.  There are no particularly well-known examples, but a list can be found here [2].  

A free-trade area is the agreement among all member countries to eliminate tariffs, import quotas, and preferences on goods and services traded between each other (the free-trade as defined in the question) [3].  Probably the best known free-trade area is NAFTA; but others are listed here and here.  

Again, there are no well-known customs unions, but a list can be found here.  

A common market has free trade and free movement of the factors of production but other economic policies of the countries may be different (one of the differences between a common market and an economic union).  You may also want to look at single market and unified market.  There are no well-known common markets, but the previous link also contains a list.  

The best known economic union is the EU though more can be found here.  An economic and monetary union has the same qualities of an economic union but each member nation also uses the same currency.  One example would be the nations in the EU who also utilize the Euro.