Trains in Europe seem like a mixture of alphabet soup and meaningless synonyms. The easiest way to sort it out is to view each train system based on the area it serves:  

• InterCity trains are long-haul trains that go from one metropolitan area to another – think of it like Chicago to New York. These trains are often nationally run trains owned and operated by a country, but are not restricted to travel within their given country.  Examples are DB (Deutsche Bahn) for Germany, Trenitalia for Italy, and SNCF for France which operates TGV INOUI, sometimes referred to as TGV, and OUIGO.
• Regional trains generally have shorter runs and serve the greater metropolitan area. They may also make longer trips to cities that are not served by the national trains. Depending on how far they travel, streetcars may or may not fit into this category. Think of this as being similar to the commuter trains serving New York city from Connecticut and Long Island. Examples of these are TER (Transport Express Régional) in France and Great Western Railway (GWR) in the UK.
• The last category of trains generally operates within the city area. Examples are the Metro in Paris, the Underground in London, U-bahn in Frankfurt.  These are similar to the subway system in New York.