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 16, where Timothy is called a "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" in his service to his "father," Paul (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 Corinthians, Paul name-drops Timothy quite a bit. He sends Timothy ahead to Corinth and asks the group to give him a warm welcome (1 Corinthians 4:17). Timothy also appears to be kind of a co-writer for the first part of 2 Corinthians (1:1). When Paul refers to "we," he probably means himself, Timothy, and other apostles from his entourage. Although who are we kidding? Paul was clearly running that show.
Overall, Timothy is usually just hanging around to shore up Paul's points. After all, it's not like Paul is the only one who thinks he's a pretty great apostle. The guy has a sweet groupie in Timothy, and this young man is clearly ready to step up and defend his boss when needed.