Author
The gospel of Matthew itself never actually mentions who wrote it. However, very early Church tradition is that Matthew, the tax-collector-turned-disciple of Jesus, is its author. Since there are no other competing traditions, this is going to be our assumption as well!
Date
Many scholars date Matthew to the late 1st Century AD. This allows for Matthew to have used the gospel of Mark as a source. Mark, which we believe is based on the teachings of Peter, was likely the earliest gospel written. A late 1st century date also makes sense of clear references in Matthew to the destruction of Jerusalem by the Romans, which happened in 70 AD.
ORIGINAL AUDIENCE
This is difficult to pin down with any accuracy, but there is a strong case to be made for Matthew's original audience being the Church community in Antioch, an important 1st century city in Syria. The city was a melting pot of different cultures and traditions, and it would have been a perfect place for Gentile believers, untrained in the Hebrew Bible, to begin flooding the church, which is a theme Matthew's gospel addresses frequently.
Antioch would have also seen fractures developing in the Jewish synagogue communities as some Messiah-believers were expelled and persecuted for their devotion to Jesus. When you add in the certain chaos caused by the Roman destruction of Jerusalem and growing cultural opposition to "the Way," the church in Antioch would have been simmering with many of the problems Matthew's gospel is tailor-made to address.