The epistle was written by the Apostle Paul (2 Timothy 1:1).
"It was probably written a year or so after the first, and from Rome, where Paul was for a second time a prisoner, and was sent to Timothy by the hands of Tychicus." -Easton's Bible Dictionary (1897)
In this epistle Paul entreats Timothy to come to him before winter, and to bring Mark with him (compare Phil. 2:22). He was anticipating that "the time of his departure was at hand" (2 Timothy 4:6), and he exhorts his "son Timothy" to all diligence and steadfastness in the face of false teachings, with advice about combatting them with reference to the teachings of the past, and to patience under persecution (2 Timothy 1:6-15), and to a faithful discharge of all the duties of his office (2 Timothy 4:1-5), with all the solemnity of one who was about to appear before the Judge of the quick and the dead.