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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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.
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
Between regions there is the
National trains are
They compare to the trains in Tokyo as follows
And, of course, 電車 just means "train" and not any particular type of railway.