I know that they are all trains, but what would their counterparts be in the
US or even Western Europe? Or better yet, when do you use one train as opposed
to the other?
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words
What's the equivalent to densha, futsuu, tokkyuu, and kyuukou in the West?
電車 【でんしゃ】 (densha) is an electric train, because 電 refers to electric.

普通 【ふつう】 (futsuu) is a local train, one that stops at every station.

特急 【とっきゅう】 (tokkyuu) is an express or limited stop train. These usually only
stop at main stations to reduce the travel time. 急 refers to the speed such as
rapid or fast.

急行 【きゅうこう】 (kyuukou) is a high speed train, relatively speaking, in any area.
Refers to the speed of the train, not necessarily the transit time. Rough
interpretation of the kanji would mean something like "moves fast".

I doubt these types are specific to any country, to the extent such services
are available.
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Because "in the West" is quite general, I'll just draw a parallel with
Germany. In Germany, the national rail network has similarities to the system
in Tokyo.

On a regional level there are the

  * [RB](http://en.wikipedia.org/wiki/Regionalbahn) ( _Regionalbahn_ "regional train"), which stops at any town with a train station
  * [RE](http://en.wikipedia.org/wiki/Regional-Express) ( _Regional-Express_ "regional express"), which leaves out the smallest stations

Between regions there is the

  * [IRE](http://en.wikipedia.org/wiki/Interregio-Express) ( _Interregional-Express_ ), which connects bordering regions and stops at large towns or cities

National trains are

  * [IC](http://en.wikipedia.org/wiki/Intercity_%28Deutsche_Bahn%29) ( _Intercity_ ), which stops only at cities
  * [ICE](http://en.wikipedia.org/wiki/Intercity-Express) ( _Intercity Express_ ) which stops only at the bigger cities

They compare to the trains in Tokyo as follows

  * RB = 各駅停車, 普通
  * RE = (通勤)快速
  * IRE = 急行
  * IC = 準特急
  * ICE = 特急

And, of course, 電車 just means "train" and not any particular type of railway.
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