This is probably the weirdest part in Ezekiel, and everyone has their own interpretation. But before we get into those, we ought to marinate in the bizarre glow of these trippy passages:
As I looked, a stormy wind came out of the north: a great cloud with brightness around it and fire flashing forth continually, and in the middle of the fire, something like gleaming amber. In the middle of it was something like four living creatures. This was their appearance: they were of human form. Each had four faces, and each of them had four wings. Their legs were straight, and the soles of their feet were like the sole of a calf's foot; and they sparkled like burnished bronze. (1:4-7)
Under their wings on their four sides they had human hands. And the four had their faces and their wings thus: their wings touched one another; each of them moved straight ahead, without turning as they moved. As for the appearance of their faces: the four had the face of a human being, the face of a lion on the right side, the face of an ox on the left side, and the face of an eagle; such were their faces. Their wings were spread out above; each creature had two wings, each of which touched the wing of another, while two covered their bodies. (1:8-11)
Now, this vision wasn't a completely out-there addition to Ancient Near Eastern literature. Cherubim, combining the features of different animals, show up in Ancient Near Eastern Art (especially Babylonian art, which Ezekiel might've been exposed to during the exile) a lot, as guardians of royal temples and other spots of importance.
Everyone's a Critic
Everyone has their own take on these weird visions.
John Calvin thought that the animals were chosen to pull God's chariot just because they were the kings of their respective parts of the animal kingdom—the Human (humanity), the Lion (mammals), the Eagle (birds), and the Ox (…the Ox is a king, too, we guess?). But for this analysis, we need to get farther out than that.
The early Church Father, Irenaeus, thought that they each represented one of the Gospel writers who helped usher Jesus' message around the globe. The Human represents Matthew, since Matthew focuses on Jesus' human lineage and genealogy at the beginning; the Ox stands for Luke, since the Ox is a work-animal, and Luke talks about the physical acts and healing miracles of Jesus; the Lion is Mark, because Mark discusses Jesus as a figure of royal stature; and the Eagle is John, who is able to fly into the spiritual mysteries of Jesus' identity as "The Word."
On the Jewish side of things, the Mishnah (an ancient rabbinic text composed around the 2nd and 3rd century CE) announced that most people should steer super clear of this passage, unless they're really really far along in their spiritual training. Some combination of theological sensitivity (describing the literal, physical God is always a hot-button issue) and fear of misinterpretation (not everyone has the subtlety of mind/sanity to deal appropriately with psychedelic prophecies) led one rabbi to announce that nobody should read it...ever. But, of course, people did, and the mystical souls amongst them got a ton of mileage out of this outlandish vision of the heavenly landscape. This vision was a pretty big deal to the medieval kabbalists (source).
For the 18th-19th-century British poet and prophet, William Blake, God and humanity are one. So Ezekiel's four animals represent the divine aspects of both God and of human beings. The Human is Imagination; the Lion is Passion, Love, and Desire; the Ox is Power and Sensation; and the Eagle is Intellect.
Of course, there are other interpretations, but these should be enough to chew on for now.