These four beasts, from Daniel's apocalyptic vision in chapter 7, are the same as the four Empires represented in the statue Nebuchadnezzar sees in his dream, except that dream was actually a relatively staid and buttoned-down one, compared to this vision. Here, the four beasts arise out of the sea, symbolizing (maybe) chaos.
The first is a lion with eagles' wings, who loses those wings, stands up like a human, and receives a human mind—apparently because the lion is Babylon. This reflects the part of Nebuchadnezzar's transformation where he goes from being like an animal back to being a human. This suggests that the Babylonian Empire and its leaders were, to a certain degree, improved and were made more humane over time. You may remember that the Babylonians also took the gold in the statue analogy—this Daniel guy is consistent.
After the Babylonians come the Medes, probably the most "whatever" of these empires, despite the fact that they get to be a pretty cool bear who chews on three tusks (or ribs) and looks fearsome and murderous. The Persian Empire shows up in leopard form, like in some zany Animorphs novel (before your time, we know), who has four wings and four heads, probably representing the four Persian kings.
Heavy Metal Monster
The last beast represents the Greek Empire—which was led by Alexander the Great, at first, and conquered a massive empire bigger than all the rest. It was also the empire that the author of the Book of Daniel (remember that it was probably completed around 165 BCE, when the Greeks were all squabbling over succession, still making sure that somebody had a hand free with which to oppress the Israelites) had the most personal experience with—so, it gets pretty evocative. That's why this beast is so weird and different from all the others—stamping around, crushing stuff, gnashing its iron teeth: "It had great iron teeth and was devouring, breaking in pieces, and stamping what was left with its feet. It was different from all the beasts that preceded it, and it had ten horns" (Dan 7:7).
Specifically, it represents the Seleucid Empire, which was led by one of Alexander's generals and encompassed most of Mesopotamia and Syria. It has ten horns representing the ten kings of the Seleucid Empire, while the little horn represents notorious bad boy Antiochus IV Epiphanes, who kicks out three of the earlier horns because he was the fourth in the line of succession after three of his relatives.
In a way, these beasts anticipate some of the big, bad beasts of Revelation. But whereas Revelation requires more work to interpret, in Daniel's case the beasts are direct and pointed references to specific Empires. With the general impression, of course, that they're all very beastly.
The Ram and the Male Goat
After going through all four of the Empires in the earlier beast vision, Daniel focuses on two of them who were particularly important in the history of the Jews: the Persian-Median Empire and the Greek or Hellenistic Empire. (A ram and a male goat are, we're sorry to say, significantly less intimidating than the beasts of the first vision.)
The two horns of the ram represent the Medes and the Persians, while the one big horn of the male goat—who shows up to beat the tar out of the ram—is none other than Alexander the Great, himself. When the big horn gets smashed at the height of the male goat's power, that's Alexander dying (he died really young). The four horns that then pop out of the goat's head are Alexander's four generals, who divide the Empire between them.
The little horn that then pops out from one of the four horns symbolizes Antiochus IV Epiphanes (just like it did before: A+ for consistency, Dan), the secret villain of these visions. His transgressions aren't just against the Jewish people—preventing them from giving their burnt offerings to God—but against heaven itself. He tramples on angels and stars that he knocks out of the sky. Fortunately, Daniel hears two angels saying that this reign of wickedness will be limited in time. Antiochus is going to get his comeuppance when his appointed days are up.