Starting with the Battle of Loum and the One Week War, the battle progressed from space, to Earth, and back to space again as the Federation pushed the Zeon forces back into their homeland of Side 3. The final crushing blow was dealt to Zeon at the battle of A Baoa Qu, not simply because they had lost the majority of their military forces, but also because the last two members of the Zabi family in power (excluding Mineva, infant daughter of the deceased Dozle Zabi, and her mother) were killed in the course of the battle. Shortly after, Zeon's Prime Minister Darcia signed a peace treaty to end the war.
Some of the more notable battles of the One Year War were Operation Odessa, in which the Federation mounted a large scale assault on Zeon's precious Odessa Mining base. The Federation's victory at Odessa effectively removed Zeon from the Earth, and the last main remnants of Zeon were finished off at the battle for California base, the last Zeon stronghold on Earth. Additionally, the Battle of Loum, which began the war, was a crushing defeat for the Federation, though the colony Zeon had intended to drop on the Federation's Earth HQ at Jaburo ended up veering off course and landing in Australia. Later in the war, the attempted capture of Jaburo via a drop operation led by Commander Char Aznable ended in failure for Zeon once again. The Battle of Solomon was one of the last great battles in the war, and the Federation succeeded in capturing Zeon's secondary asteroid base.
Notable pilots include Amuro Ray, pilot of the Gundam; Char Aznable (Casval Deikun), ace Zeon pilot, "The Red Comet", and Amuro's rival throughout the various UC series (except, perhaps, for Zeta Gundam, in which they fought side by side against the Titans); Johnny Ridden, another Zeon ace, nicknamed "The Crimson Lightning"; Shin Matsunaga, yet another Zeon ace, nicknamed the "White Wolf (of Solomon)"; Shiro Amada, a Federation pilot who fell deeply in love with Aina Sahalin, a Zeon officer's daughter, and consequently left the Earth Federal Forces; Yuu Kajima, pilot of the Federation's prototype Blue Destiny Units 1 and 3; Kai Shiden, pilot of the White Base's Guncannon; Hayato Kobayashi, pilot of the White Base's Guntank (and also Guncannon, the the Mobile Suit Gundam movies); Gaia, Mash, and Ortega, also known as the "Black Tristars", a Zeon ace mobile suit unit that was destroyed by Amuro around the time of Operation Odessa; Ramba Ral, a Zeon ace also defeated by Amuro some time before the Tristars; Sayla Mass (Artesia Deikun), Char's sister, and pilot of the White Base's core booster/G-Fighter; Sleggar Law, pilot of the White Base's core booster/G-fighter in the last half of the series, died fighting Dozle Zabi's Big Zam; and Anavel Gato, a Zeon ace nicknamed "The Nightmare of Solomon".
If you want the short version, straight from GundamOfficial.com, it is as follows:
"The devastating war waged between the Earth Federation and the Principality of Zeon during the year U.C. 0079. Since it begins on January 3 and ends on December 31 of this year, the conflict is later called the One Year War. This is the setting of several stories in the Gundam saga, including the original Mobile Suit Gundam, The 08th MS Team, and Gundam 0080."