Some suggest Paul as the author, but the author does not name himself, which is not typical for Paul. The Eastern Church lists it among Paul's writings. Tertullian suggests Barnabas as the author. Several early letters and works quote it, which shows it was considered part of the authoritative Christian writings by the second century.
Whoever the author is, he was apparently a second-generation Jewish Christian who quotes from the Greek translation of the OT called the Septuagint. He makes references to ancient tabernacle procedures and not current temple rituals. He writes using classical Greek grammar and syntax (although Clement of Alexandria says that Paul wrote the letter in Hebrew and it was translated into Greek by Luke.)
Hebrews was written before the fall of Jerusalem and the destruction of the Temple in 70 A.D. The readers are experiencing some persecution which might fit into Nero's time. It may also refer to the revival of rabbinical Judaism in the late first century.
Some of the letter is clearly written to discourage "re-Judaizing" the faith (bringing back circumcision and dietary laws).
Dealing with persecution also seems to be a prompt for the writing, both in the sense of enduring the persecution and acting in solidarity as the Church.
It is addressed "to the Hebrews" and by extension then to all Jews. No location is given, so unlike many of the other letters, Hebrews is not intended for a specific congregation. It seems to be directed to both Jews who were already believers in Christ and Jews who would listen to an intellectual argument for belief in Christ.
The Jewish Christians are encouraged to publically declare their fellowship with the Church as a whole, including Gentile believers. They are also called to carry the Gospel message into their world.