An army was defined as follows from H.L. Scott's 1863 "military dictionary."
The US Constitution, Article I, section 8, gives to Congress the power to raise and support "armies..."
These armies to be organized in whatever manner the Congress determines by laws. For example, during the war there was the regular army of the United States, there was the Volunteer force (the majority), there was the United States Colored Troops, a Veterans Reserve Corps, etc., and there were large forces of militia at different times placed in the actual service of the United States as such (like the 300,000 9-month militia called forth in 1862-63).
All of these armies were under the command of the President of the United States.
From Article 2, Section 2, the President the commander-in-chief of the army and navy of the United States, and of the militia in the actual service of the United States.
The President's lawful powers were managed through the War Department, under the Secretary of War, which organized the armies into territorial/departmental and field forces commands.
During the war both sides established territorial military departments, in which a senior officer commanded. Some of these military departments had very few troops at all. Others had enough to form separate field armies, like the "Army of the Ohio," in the Department of the Ohio, or the "Army of the Cumberland" in the Department of the Cumberland.
These departmental commands and their boundaries were frequently modified by combination or separation as necessary. For example, in 1863 the Departments and armies of the Ohio, Tennessee, and Cumberland were combined into the "Military Division of the Mississippi" under General Sherman...
General Grant noted in 1864 that only the Union's Army of the Potomac was free-standing, or unconnected with a departmental command.
The field armies were principally composed of two or more army corps.
During 1863 the United States gave permanent and free-standing designations to the various corps in the several armies.
This prevented confusion, for example, between say the 1st Corps, Army of the Tennessee, and 1st Corps, Army of the Potomac. Thenceforward each corps, nation-wide, would have a permanent numerical designation, no matter to what army it was assigned.
In Sherman's force in 1864, the Armies of the Tennessee and Cumberland had two or three army corps each. The smaller Army of the Ohio had only a single corps (the 23rd), besides its army headquarters and attached units. It if were necessary, all three of the armies under Sherman could have moved and operated independently of each other.