The Republic ended following a period of intense political turmoil and war. At the height of the chaos the Republic's scheming head of government, Supreme Chancellor Palpatine, proclaimed himself Emperor.
The Empire was ruled with absolute power by Palpatine as Emperor with Darth Vader being his assistant. The Galactic Senate still existed years after the Empire's establishment, but it had little power and was eventually dissolved by Palpatine. The real power below the Emperor was in the hands of the Grand Moffs, like Tarkin and the regional governors who were eventually granted direct control of their regions.
This power was backed up by the use of state terrorism which threatened horrific retaliation to any defiance to the government. The instrument of that terror was the Imperial military, which included the Imperial Stormtroopers and a large fleet of intimidating war vehicles like Star Destroyers and All Terrain Armoured Transports (AT-ATs or Walkers) which were intended in part to spread fear as well as destroy the enemy.
In addition a key instrument was intended to be the Death Star, a moon sized space station with sufficient firepower to destroy a planet with a single shot, shown in the first Star Wars movie, Star Wars Episode IV: A New Hope.
In non-canonical works, the Empire also emphasized humanoid supremacy and other alien species like Wookies were subject to slavery. It is believed by some that Palpatine's association with non-humans like Darth Maul in Star Wars Episode I: The Phantom Menace contradicts this.
The Empire was opposed by the Rebel Alliance, a guerilla army dedicated to the defeat of the Emperor and the restoration of the Galactic Republic. This goal was nominally achieved with the death of Palpatine and the destruction of the second Death Star at the hands of the Alliance at the Battle of Endor, as depicted in the sixth Star Wars movie, Star Wars Episode VI: Return of the Jedi. However, the Empire was too large to be destroyed in one blow; for the next decade and more, the Rebels, renamed the New Republic, fought to free the galaxy from former Imperials styling themselves as independent warlords as well as Imperial loyalists such as Grand Admiral Thrawn. These battles are chronicled in the many novels licensed by Lucasfilm.
See also: Isaac Asimov's Galactic Empire Series