Daniel 7:8 "I considered the horns, and, behold, there came up among them another little horn, before whom there were three of the first horns plucked up by the roots:

A horn again is an upcoming power or kingdom.
The horn comes from the Beast of Rome.

Daniel 7:20 "And of the ten horns that were in his head, and of the other which came up, and before whom three fell;

Daniel 7:24 "And the ten horns out of this kingdom are ten kings that shall arise: and another shall rise after them; and he shall be diverse from the first, and he shall subdue three kings.

Daniel 7:8 "and, behold, in this horn were eyes like the eyes of man, and a mouth speaking great things."

In this horn were eyes like the eyes of man.
What are these eyes?
Lets look at a few scriptures:

In the day ye eat thereof, then your eyes shall be opened, and ye shall be as gods, knowing good and evil. (Gen 3:5)
Ye shall not do after all the things that we do here this day, every man whatsoever is right in his own eyes. (Deut 12:8)
Cast ye away every man the abominations of his eyes, and defile not yourselves with the idols of Egypt: I am the LORD your God. (Ezek 20:1)

Daniel 7:20 "even of that horn that had eyes, and a mouth that spake very great things, whose look was more stout than his fellows. 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:"

The one law dealing with time that says Remember is the one most Christians want to forget.

Daniel 7:12 "As concerning the rest of the beasts, they had their dominion taken away: yet their lives were prolonged for a season and time."

Even though the previous beasts were taken over, many of their traditions were brought over into Christianity.