Meitetsu is one of the largest private railway companies in Japan, non-JR group which is operating in the Nagoya metropolitan area.
Actually the company's official name is Nagoya Railroad Co. (Japanese - Nagoya Tetsudo), almost no one uses this official name even in Nagoya. Its subsidiaries include Meitetsu Department Store and Meitetsu Inn whose official names are not Nagoya department store nor Nagoya Inn.
Meitetsu is the only railway company which runs to Nagoya's gateway, Chubu International airport - NGO. It is interesting that other major international airports; Haneda, Narita and Kansai airports are all connected by two different railway companies which one of them are belonging to JR group which means you can use the Japan Rail Pass.
Meitetsu's hub station is Meitetsu Nagoya station which is adjacent to JR Nagoya station. If you are going to Inuyama castle, take the Meitetsu Inuyama line.