Nishitetsu, abbreviation of Nishi-Nippon Tetsudo is a largest private railway company in Kyushu island, it is the only major railway company not located in Honshu island to be listed in Tokyo Stock Exchange First section. It also owned a professional baseball team Nishitetsu Lions till 70s.
Nishitetsu Fukuoka(Tenjin) station is the main terminal of Nishitetsu located in Tenjin district, the largest commercial district in Kyushu island and it is the most western located major railway station in Japan. It was opened as Fukuoka station in 1924 and then renamed to Kyushu Fukuoka in 1935. Again renamed to Nishitetsu Fukuoka in 1942 and eventually changed to Nishitetsu Fukuoka(Tenjin). I assume that Tenjin was added due to Fukuoka City Subway station which is adjacent to NIshitetsu Fukuoka(Tenjin) is called not Fukuoka station but Tenjin station.
Please be aware that Nishitetsu Fukuoka(Tenjin) is not located next to JR Hakata station but take Fukuoka City Subway to travel each other. Nishitetsu Fukuoka(Tenjin) station is a gateway of tourists visiting Dazaifu Tenmangu shrine in Dazaifu city. JR pass cannot be used but SUICA is available to be used.