This, ladies and gentlemen, is free verse. Eliot, after Walt Whitman and along with poets Ezra Pound and William Carlos Williams, was one of the pioneers of the kind of freewheeling lines that we associate with so much of contemporary poetry.
But! In the words of Robert Frost, "poetry without form is like playing tennis without a net," and for the Modernist poets, this certainly held true. So while there's no stanza pattern at work here, and the lines vary a bit in terms of length, let's take a closer look at rhythm. Sure, Eliot is playing pretty fast and loose with form here, but the ghost of metrical pattern is always lurking in his lines. And that means we should be keeping a weather eye (or ear) out for a sneaky little bugger called the iamb.
An iamb is just a set of two syllables in which the second one is stressed. Think "away," or "again," or "for real." An iambic phrase would go a like this: "the little fox was red." Now, "Journey of the Magi" is by no means entirely iambic. From what we can tell, Eliot's all over the place with his stress patterns. But the key here is that certain important bits of the poem are composed in loose iambs. Check it out:
We could do this with a couple of other rhythmic feet (like trochees, or anapests) as well. But the point is that the lines aren't entirely helter-skelter. Eliot's arranged them carefully to produce driving rhythms when he wants them, while still not sacrificing the common speech feel of the poem. So maybe free verse isn't so free after all.