In this chapter we are introduced to two beasts. They are allies and both are anti-Christian and get their authority from Satan. They are blasphemous toward God.
Note the chain of command….The first beast from the sea gets its power, throne, and authority from the dragon, Satan (see 13:4, 12:9). The second beast from the land exercises the authority of the first beast – kind of like an institution created to enforce worship of the first beast.
Satan > Beast From The Sea > Beast from the Earth
Note also their goal – making war with the saints.
Interestingly, the time duration of the authority given to the first beast is 42 months, the same we saw earlier when the Gentiles were treading the holy city and the witnesses were prophesying and God’s people were protected (11:2, 3; 12:6, 14). What we are looking at in these beasts is likely a closer inspection of what was revealed earlier. In fact, this is building on what we saw about the dragon at the end of Chapter 12 “And the dragon (having failed to defeat Jesus or Michael) was enraged with the woman, and he went to make war with the rest of her offspring, who keep the commandments of God and have the testimony of Jesus Christ. These beasts are his agents in his war against believers.
The idea of a mark is something that indicated a person had worshiped the beast. Those who did not were killed or excluded from society.
The number 666 is likely some cryptic code identifying the beast. There are many theories which you can read about.
Polycarp was a leader in the church at Symrna (one of the 7 to which the book of Revelation was written) and he died as a martyr in 155 ad. Many Christians were being put to death at the time and now they seized one of the leaders. According to tradition, before he died he was given a chance to save himself by cursing Christ, saying Ceasar is Lord, or saying ‘Down with the Atheists!’”. (Atheists in this case would be Christians who denied the diety of the emperor and who believed in a false God (to them), Jesus.) Polycarp famous reply was “86 years have I have served him, and he has done me no wrong. How can I blaspheme my King and my Savior?” after which he was burned to death.
How does this account fit what we see in Revelation 13 and maybe help us understand the book?