Which civil war battle was fought from the 15th to the 16th of december 1864?

The Battle of Nashville.

Explanation:

The Confederate forces, commanded by John Bell Hood, had already suffered heavy casualties against General John Scofield, at Franklin, Tennessee. When Scofield moved north to Nashville to reunite with General George Thomas, Hood followed him, despite being greatly outnumbered, and placed his troops outside Nashville's strong defenses.

Thomas saw the opportunity to strike the decisive blow and took it. The confederate army suffered a terrible defeat, and the few survivors had to retreat to Mississippi.

Hood signed his resignation shortly after that.