The remnants of the Galactic Empire were indeed powerful, but they were fragmented and splintered under leaders such as Warlord Zsinj, Grand Vizier Pestage, and Director of Intelligence Ysanne Isard. Infighting among them crippled their response to the Alliance/New Republic's military advances. Only under Grand Admiral Thrawn was the Empire reunited to any significant extent in the years immediately following the destruction of the Death Star and the death of the Emperor.
So, yes, there was another war, but it was really a continuation of the same war between the Alliance and the Empire that had begun even before the events of Episode IV.
One of the decisive moments in the war was the Liberation of Coruscant. This was detailed in the novel X-Wing: Wedge's Gamble. Under Wedge Antilles, Rogue Squadron sneaked on-planet and sabotaged the powerful defense shield. Additionally,
Sixteen of the galaxy's worst criminals were also freed from the prison colony of Kessel and loosed on the Imperial City so that they could create a diversion to distract the Empire. The Rebels would later regret this; these criminals would go on to resurrect the Black Sun organization
Then, Admiral Ackbar entered the system with a fleet and overwhelmed the limited defenses there. Since the planetary shield was so powerful, only a few Star Destroyers were on station at any one time, due to the limited amount of capital ships available to the remaining Imperials. The ground fighting, however, was not so easy:
Since any large ground invasion force would be detected by the Imperial sensor net, General Brenn Tantor landed a very small initial ground force near Coruscant's north pole, which proceeded to capture Imperial command centers and disable sensor nodes. This allowed Tantor to bring down a larger force undetected, which he used to pound away at the defenses between himself and the Imperial Palace. However, the city planet would not be taken easily, as forces under Grand General Malcor Brashin continued to resist the Republic invasion force, leading to intense building-to-building warfare.
After the fighting, which for the size and importance of Coruscant, was limited, the New Republic took command. However, in the aftermath, the Imperials unleased the Krytos virus on the alien population, an event that almost tore the New Republic apart.
After this (which took place in 7 ABY, though sources prior to the release of Wedge's Gamble placed it in 8 ABY), the Imperials retook Coruscant in 10 ABY, and held it for only a year until the New Republic took it again for good; at least until the events of the Yuuzhan Vong war.