The Rebel Alliance was created to oppose the Empire. The war began after the Rebel Alliance declared its intentions. Emperor Palpatine was pleased with a rebellion against his rule, because it would give him an excuse to get rid of the last remains of the Republic. A fringe benefit was that he could display the power of the Imperial Navy. This confidence in his regime's power became his greatest weakness.
The Rebel Alliance was not very successful in its first year. But the Battle of Yavin(0 BY), which resulted in the destuction of Death Star, marked a turning point in the war and Galactic History. The Empire struck back (in the Empire Strikes Back) and began building a new Death Star. This Death Star was destroyed in the Battle of Endor(4 ABY), which also resulted in the death of Palpatine and his right-hand man, Darth Vader.
Please note that all information that follows is from the Expanded Universe.
The Empire fell into warlordism, but it wasn't dead. The Rebel Alliance began trying to take back Coruscant and create a New Republic. They eventually succeeded. However, The Empire rose again several times under powerful leaders such as Grand Admiral Thrawn and a series of clones of Palpatine.
When a peace treaty was signed between the tiny fraction of the galaxy that was under the control of the Empire and the New Republic(20 ABY), the Galactic Civil War ended.