In the book of Hebrews, the author doesn't identify himself (contrary to Paul's usual practice) but he was obviously well known to the original recipients. It is clear in the letter that the writer must have had authority in the apostolic church, was intellectual, and was well versed in the Old Testament. Barnabas met these requirements and evangelized with Paul so this is why he is thought to be a possible author.
The other leading candidate for authorship is Apollos. He was also a Jewish Christian with notable intellectual and oratorical abilities. Luke tells us that "he was a learned man, with a thorough knowledge of the Scriptures" (Ac 18:24). We also know that Apollos was associated with Paul in the early years of the church in Corinth.
Two things are evident: The author was a master of the Greek language of his day, and he was thoroughly acquainted with the Greek translation of the Old Testament, which he regularly quotes.
The letter was addressed primarily to Jewish converts who were familiar with the OT and who were being tempted to revert to Judaism or to Judaize the gospel. Judaizers are Christians who teach it is necessary to adopt Jewish customs and practices, especially those found in the Law of Moses, to be saved.
The theme of this book is to point out the absolute supremacy and sufficiency of Jesus Christ a revealer and mediator of God's grace. The first few verses outline Christ as God's full and final revelation, far surpassing the revelation given in the OT. The prophecies and promises of the OT are fulled in the "new covenant" of which Christ is the mediator.
The Jewish Christians (as well as us) are told that there can be no turning back to the old Jewish system, which has been superseded by the unique priesthood of Christ.