This dude is Paul's companion. A co-disciple. His bosom buddy.
Timothy is mentioned several times throughout Paul's letters so it's clear that the two men had a long working relationship. They first meet in Acts of the Apostles 16, where Timothy is called and "disciple" and Paul arranges for him to be circumcised. Well, that's a fine how-do-you-do.
Throughout their lives, Paul probably functioned as kind of a mentor for Timothy. He sends him all around the Roman Empire visiting various churches. He praises Timothy repeatedly saying, "I have no one like him," and that the kid is "like a son" (Philippians 2:20, 22). Timmy even gets two books of the Bible named after him—The First and Second Epistles to Timothy. Of course, Paul probably didn't write either one. But it's still impressive, right?
In both letters to the Thessalonians, Timothy's name is dropped quite a bit. Paul sends Timothy to Thessalonica when Paul can't go back himself (1 Thessalonians 3:2). The kid also appears as a kind of co-writer at the beginning of both Thessalonian epistles. When Paul refers to "we," he probably means himself, Timothy, and other apostles from his entourage. Even though…who are we kidding? Paul was really running that show.
Overall, Timothy is usually just hanging around to help Paul spread the message of Jesus around the Roman Empire. He teaches in Paul's absence (1 Thessalonians 3:2) and reports back to Paul on all the comings and going in the congregation (1 Thessalonians 3:6). Lucky for the Thessalonians, he has a good report. Timothy is not above ratting out a fellow Christian if that's what the boss man wants.