To understand the book of Revelation you need to be very familiar with the books of Daniel and Ezekiel.
Daniel shows us that empires or civil powers are pictured as beasts. Daniel 7:17, “These great beasts, which are four, are four kings, which shall arise out of the earth.” This representation is still a common practice today, as the USA pictures itself by the bald-eagle and Russia as a bear.
Now, we must compare Revelation 13 to Daniel 7 to find the identity of the 10 horned beast in Daniel.
Revelation 13:1, “And I stood upon the sand of the sea, and saw a beast rise up out of the sea, having seven heads and ten horns, and upon his horns ten crowns, and upon his heads the name of blasphemy.”
Daniel 7:7, “After this I saw in the night visions, and behold a fourth beast, dreadful and terrible, and strong exceedingly; and it had great iron teeth: it devoured and brake in pieces, and stamped the residue with the feet of it: and it was diverse from all the beasts that were before it; and it had ten horns.” Therefore, both Daniel and Revelation are describing the same beast.
Revelation 13:5, “And there was given unto him a mouth speaking great things and blasphemies; and power was given unto him to continue forty and two months.” (1260 days) Revelation 13:7, “And it was given unto him to make war with the saints, and to overcome them: and power was given him over all kindreds, and tongues, and nations.
Daniel 7:21, “I beheld, and the same horn made war with the saints, and prevailed against them;”. Daniel 7:25, “And he shall speak great words against the most High, and shall wear out the saints of the most High, and think to change times and laws: and they shall be given into his hand until a time and times and the dividing of time. (1260 days) In both contexts, the beast blasphemes God and persecutes the saints for 1260 days.
History shows the fourth universal empire was Rome. When we look at the image, we see there is iron mixed with clay. This pictures the Pagan Roman Empire transitioning into the Holy Roman Empire—that is the Papacy.
The other beast is a little harder to identify. The 2 horned beast can do many signs and wonders in the future and will cause the world to worship the Papal beast. This beast or civil power is a lot like the Papal beast and even makes an image of the Papal Beast which it also gives power to.
Revelation is a history of the Christian church throughout the Gospel Age and though Islam is very much like the Papal beast in that it is both church and state, we don’t think Islam is pictured here in the book of Revelation.
We see the Image of the Beast as the creation of a false Christian church-civil government union. This union will exercise great power such that any who refuse to honor that system will be persecuted and killed. (However, this is another topic which requires more space to explain than provided for this question.)