In the TV anime Gundam (Universal Century) universe, Zeon can refer to either: Republic of Zeon, Principality of Zeon or Neo-Zeon.
The Republic of Zeon was initially founded by Zeon (Jion) Zum Daikun, who declared the colony located at the second Earth-Luna Lagrangian Point, named Side 3, independent from the Earth Federation. After Zeon's (suspicious) death, his advisor, Degin Sodo Zabi becomes leader and renames it the Principality of Zeon. War is declared on the Earth Federation shortly thereafter.
The result is the One-Year War. The first week, known as the One-Week War (Jan 3 - Jan 10, UC0079) resulted in more than a quarter of the Earth's population of 12 billion people perishing, especially residents of the Side 5 colonies. The Earth Federation and the Principality of Zeon signed the Antarctic Treaty (Jan 31) to prevent nuclear warhead use and colony drops as a result of the astounding losses.
The very next day (Feb 1) after the treaty, Zeon begins the formation of its Earth Attack Force. By the middle of 0079, the Zeon forces had control of about 70% of the Earth. Nevertheless, after several major operations (Velvet, Odessa, et al.), the Earth Federation regained much of the lost ground. The war eventually climaxed in the Battles of Solomon and Abawaku (A Baoa Qu), finally resulting in the end of the war. The final surrender occurred with the Treaty of Granada on January 14, UC0080.
Several years later (UC0083), the Delaz fleet, under Aguille Delaz, sought to rebuild Zeon in Operation Stardust and the resulting colony drop destroyed much of the North American farmland. The Delaz fleet was soon defeated, but the war led to the Earth Federation's creation of a force called the Titans to prevent future insurrection.
Several more years passed as the Titans reigned with brute force. In UC0088 the Titans were eventually defeated by the Anti-Earth Federation. Afterwards, seeing this opportunity, a new Zeon arose under the leadership of Zeon Daikun's son, Cassaval Rem Daikun (aka Char Aznable, the Red Comet). Neo-Zeon was founded under the goal of moving all of humanity into space so the Earth could be a sanctuary. Char's plans involved colliding the huge asteroid, named Axis, onto the Earth, thus making it uninhabitable, but his plan was foiled by his old adversary from the One Year War, Amuro Ray, and Neo-Zeon was quickly crushed.
In UC0100, The Republic of Zeon officially rejoined the Earth Federation.
Early official romanizations were rendered "Zion," the creators of Gundam apparantly being unaware of that word's connotations. Early fansubbers often used the alternate romanization "Jion." Frederick Schodt chose to use "Zeon" when he translated the Gundam novels into English, and this later became the official romanization.