Short answers, pulled from the story.
The Army of the Cumberland was originally called the Army of the Ohio. It was created in November 1861 under Brigadier General Robert Anderson, and was renamed when Major General William S. Rosecrans took command and merged the army with the Department of the Cumberland in October 1862.
The Army of the Cumberland had two primary commanders. Major General William S. Rosecrans led the army from the 24th of October 1862 until the 19th of October 1863, overseeing the battles of Stones River, the Tullahoma Campaign, and Chickamauga. Major General George H. Thomas then commanded the army from October 1863 through August 1865, leading it through Chattanooga, the Atlanta Campaign, and Nashville.
Rosecrans was replaced because of his defeat at the Battle of Chickamauga and his inability to break the Confederate siege of Chattanooga. Ulysses S. Grant arrived at Chattanooga on the 19th of October 1863 and chose George H. Thomas as his replacement, despite Rosecrans having been a popular and respected commander.
The Army of the Cumberland was assigned only to seize the rifle pits at the base of Missionary Ridge during the Battles for Chattanooga in 1863. After completing that limited objective, four divisions spontaneously stormed up the ridge and routed the Confederate center without orders. When Grant demanded to know who authorized the charge, both Thomas and corps commander Gordon Granger said they did not know.
The Army of the Cumberland was a central component of Sherman's army group that captured Atlanta in September 1864. After Atlanta fell, part of the army under Thomas defeated Confederate General John B. Hood at the Battle of Nashville, while the XIV and XX Corps marched with Sherman to the sea and through the Carolinas under Major General Henry W. Slocum, designated the Army of Georgia.
The Grand Review of the Armies was a victory parade held in Washington, D.C., before President Andrew Johnson in 1865. Elements of the Army of the Cumberland that had marched with Sherman, redesignated as the Army of Georgia under Major General Henry W. Slocum, participated in the review.