Both letters to Timothy, along with the letter to Titus, are known as the “pastoral epistles.” All were written by an anonymous author (or authors) in the Pauline tradition, and they were accepted by the church as such. Most scholars date them very late, around 100 AD.
The purpose of all the pastoral epistles was to exhort shepherds or pastors about church order. They are like a memo to a subordinate official on the assumption of a new position. They describe duties and prerogatives, as well as remind the official of their important qualifications.
The notion of a church hierarchy took time to develop. By the late first century, there was some sense of “order” but there was little consistency on titles and there was certainly no org chart. For Paul, the notion of “church” (ekklesia) was the body of Christ. By the time of the pastoral epistles, the ekklesia was seen as the assembly of Christians who were properly overseen by shepherds in the interest of preserving sound teaching. The tension between Paul’s vision and the development of that into the hierarchy is a perennial tension still felt today.
The letters are addressed to Timothy and Titus, two of Paul’s closest companions. Timothy is referenced in Acts 16:1-3, 17:14-15, 18:5, 19:22, and 20:4, Romans 16:21, 1 Corinthians 4:17, 16:10, 2 Corinthians 1:1, 1:19, Philippians 1:1, 2:19, Colossians 1:1, 1 Thessalonians 1:1, 3:2-6, 2 Thessalonians 1:1, Philemon 1, and Hebrews 13:23. Titus is mentioned in 2 Corinthians 2:13, 7:6-14, 8:6-23, and 12:18, Galatians 2:1-5, and 2 Timothy 4:10.
The present grouping of these letters is probably based on number of words, from greater to lesser. Titus very likely originally served as an introduction to the group. It contains one of the longest greeting sections in the New Testament. 2 Timothy is a sort of spiritual “last will and testament” and likely was meant to come last. You might want to read these letters in this order to get a sense of the full narrative arc: Titus – 1 Timothy – 2 Timothy.