In Europe, you have taken Euro City or Inter City trains. These trains are express trains which stop at major cities only and sometimes there are the fastest and the most comfortable trains in certain countries.
On the other hand, Germany has ICE and France has TGV which are the fastest category and you need to pay supplement to board those trains. In Japan, the fastest is Shinkansen and limited express(Tokkyu in Japanese) is the counterpart of EC/IC.
Literally Tokkyu is a shorten word for Tokubetsu kyuko which means special express in Japanese. Therefore, Tokkyu should be translated to special express but limited express is used for English translation.
Limited express ticket is charged to board limited express in addition to ordinary fare except in several private railway operators such as Tokyu and Hankyu. These rail operators which do not charge limited express ticket uses same trains which are used for local and other type of slower trains. Which means limited express with limited express tickets use higher class trains usually seat reservations are compulsory or cars with seat reservation is needed.
In Japan, each limited express have its distinguished name and followed by number. Azusa No.1 means limited express is called Azusa and it is the first Azusa to depart on that day, followed by Azusa No.3 and then Azusa No.5. Odd numbers are outbound trains; for example departing from Shinjuku station in Tokyo. On the other hand, even numbers are inbound trains heading to Shinjuku station.