Heisei Era refers to the timeline in the original Fushigi Yuugi series. This is the era when Miaka and Yui went into the Shijintenshisho and summoning their respective Gods Suzaku and Seiryuu. This is also the era when the Priestess of Byakko reunited with Tatara in the afterlife and her story being told by Toki to Keisuke and Tetsuya.
Heisei era started in December 1989 and is still ongoing as of 2015. It's named after Emperor Akihito, grandson of Emperor Taishou (from the era where Fushigi Yuugi Genbu Kaiden took place) and son of Emperor Showa (from the times where Fushigi Yuugi Byakko Ibun happened). It's marked by both the rapid growth of Japanese economy as well as its military regrowth, but also by the problems coming from recession, political crisis and several earthquakes.