Heisei Era refers to the timeline in the original Fushigi Yuugi series. This is the era when Miaka and Yui went into the Universe of the Four Gods and summoning their respective Gods, Suzaku and Seiryuu, and some time later Mayo Sakaki was almost tricked into summoning a fake Suzaku. This is also the era when the Priestess of Byakko reunited with her lover Tatara in the afterlife, and her story is told to Miaka's brother Keisuke and his friend Tetsuya (offscreen in the manga, onscreen and by her grandson Toki in the anime series). If one counts the video games, it's also when other three young girls (Madoka Ootori, Misaki Himuro and Mariko Kobayashi) had their own adventures.
The Heisei era started in December 1989 and is still ongoing as of 2018. It's named after Emperor Akihito, grandson of Emperor Yoshihito/Taisho (from the days where Fushigi Yuugi Genbu Kaiden, Fushigi Yuugi Byakko Ibun and the very first part of Fushigi Yuugi Byakko Senki took place) and son of Emperor Hirohito/Showa (from when Fushigi Yuugi Byakko Senki properly happened). It's marked by both the rapid growth of Japanese economy as well as its military regrowth (as the JSDF alias Japan Self-Defense Dorces), but also by the problems coming from recession, several socio-political crisis through decades, and many earthquakes.
The original Fushigi Yuugi manga began its run in 1992 and finished in 1996, so it can be assumed that Miaka and Yui's adventures took place in a similar time frame.