Both characters debuted in Final Fantasy VII. In the game Aerith was a major character and a party member. Zack was a minor and hidden in the game. His story has since been expanded on in as sequels and prequels to FFVII have came out.
Zack Fair grew up in a backwater town. At a young age he became endeared with SOLDIER, a superhuman organization and a branch of the military of the Shinra power company. He travels to the city of Midgar to apply and is admired to SOLDIER. The talented young man quickly raises to 2nd Class.
Aerith was the daughter of Ifalna, the last of the Cetra, and Gast, head of the Shinra science department. However Gast was killed by Hojo and mortally Ifalna wounded. Ifalna entrusted Aerith's life to Elmyra Gainsborough who raised her as her daughter in Midgar. Growing up, Aerith took a liking to a local Church and spent time growing flowers there.
One day while on a mission form Shinra Zack falls off the upper plate of the city, crashing through the roof of the church, startling Aerith. After the first impression, Aerith agrees to a date with Zack. The two spend a lot of time thereafter and become boyfriend and girlfriend. Two years later, Zack is on a mission to Nibelheim when things go from bad to worse and Shinra declares him legally dead. Zack is severely wounded after the Nibelheim Incident and is captured and experimented on by Hojo for over four years. Aerith never learns what happened. She writes letters to Zack hopping she could reach him. After writing 89 letters, she loses hope of ever seeing Zack again and has to let go.
Zack eventually escapes with his comatose friend Clould Strife. Now an enemy of Shinra, who is trying to coverup the truth of what happened in Nibelheim, he spends months on the run trying to reach Midgar. On the outskirts of the city Zack is gunned down and killed by the Shinra Army but Cloud survives and travels to Midgar. Cloud eventually meets Aerith and helps rescue her after she is captured by Shinra.
Eventually Aerith is slain by Sephorith while trying save the planet. Aerith never realizes that her ex-boyfriend is dead or that his death is part of the reason behind Cloud particular behaviors (or least while she is alive). Aerith always referred to Zack as her ex-boyfriend when asked but never gives his name. She mentions to Cloud that he reminds her of Zack and the is part of the reason why she felt attracted to him. Aerith and Zack appear occasionally in the Lifestream together.
Zerith is a popular ship in the Final Fantasy fandom. One of the advantages of the ship is the fact both are deceased by the endgame of the canon story-line. The ship has supporters from the extremely popular CoTi ship and many CoTi fans ship Zerith since it does not conflict with CoTI. Even if no one but CoTI fans shipped Zerith, the sheer number of CoTi fans would make it one of the more popular ships in the Final Fantasy fandom.
Many Zerith fans feel Zack and Aerith personalities complement each other. Some are drawn to the tragedy of the ship. Both Zack and Aerith deaths are considered to be the among saddest moments in the Final Fantasy franchise. Their tragic fate is contrasted with the perky energy of pair exhibit. The two are optimists in a world of cynics.