The Wutai War was a military conflict that took place several years before the start of Final Fantasy VII, ending at the beginning of Crisis Core -Final Fantasy VII-. It was a conflict between the modern power of Shinra Electric Power Company against the fiercely independent western nation of Wutai. The war takes place from sometime around 1992 until 0001.
The initial spark was over a dispute of a Mako Reactor. Shinra wished to place a reactor at Wutai, and thus economically dominate the nation like the rest of the world. Wutai, lead by Godo Kisaragi, refused to surrender to Shinra authority, starting a long conflict.
Shinra held a clear advantage over Wutai as its power dominated the entire Planet. They held the world's most advanced technology, having planes, machine guns, robots, and the genetically enhanced special forces unit: SOLDIER. Sephiroth, still in SOLDIER at the time, became world famous as the hero of the war, inspiring many young men to dream of joining the organization to become a hero like him. Shinra conscripted forces from around the world, grabbing Elmyra Gainsborough's husband as a regular from the city of Midgar.
The final battle of the Wutai War took place at Fort Tamblin, the last major place of resistance for Wutai's ninja army. A young SOLDIER 2nd Class, Zack Fair, took the fortress single-handedly with his mentor, Angeal Hewley, standing by. Whatever remained of Wutai's resistance is defeated in the Crisis Core missions, To End the War with Wutai.
Godo Kisaragi, after putting up so much resistance, surrendered the control of his homeland to Shinra. To show his surrender, he banned the use of Materia. Wutai, once the last bastion of resistance against Shinra, was reduced to a tourist trap. Yuffie Kisaragi, disgusted with her father's capitulation, left Wutai to search the world for Materia that could save her homeland. Shinra became the unquestioned ruler of the Planet.
It can be assumed that due to the war's long tenure, both sides took heavy casualties. Elmyra Gainsborough's husband never returned home from the war, and as Elmyra's adopted daughter, Aerith Gainsborough, indicates, he died.