Theophilus was a bishop of Antioch, Paul's home church in Scripture. He's the second bishop of Antioch that we have writings from. Ignatius was the first.
He wrote a defense and explanation of Christianity called To Autolycus. It's quite long, and it covers a lot of what Christianity was to the early church. Like other long writings, there's no surprises in it. All the early Christian writers say the same things about what the apostles taught.
Theophilus also calculated the amount of time from Adam to his day, based on the birth dates in the Old Testament. He came up with 5,698 years through A.D. 168 or so. This is somewhat different from the dates we would calculate because he, like all other 2nd century Christian writers (and some NT writers), used the Septuagint, a Greek translation of the Hebrew Scriptures. Some of the birth dates are off by 100 years between the Masoretic and Septuagint texts. Seth, for example, was born when Adam was 230 years old in the LXX, rather than 130 as in the Masoretic text.
His dates create quite an interesting prophecy, though, since Rome fell in A.D. 476, very near the 6,000 year mark by Theophilus' calculations.