Let's start with the easy part: this poem is actually a kind of song, called a ballad. For the most part, ballads are conventionally written in quatrains (stanzas or verses of four lines), which are sung to musical accompaniment. A lot of our pop songs even today follow this very same pattern.
Meter-wise, the ballad is a form that comes to us from oral tradition, which accounts for the fact that the meter of "Lord Randall" is a lot less precise than some other, more modern poems you may have read. Basically, these lines are written to fall within the more flexible rhythms of a song, and not to fit within the kinds of metrical patterns we often think of when scanning poems.
That means there isn't a consistent number of syllables per line, nor is there a clear pattern to the stresses and unstresses in each line. Basically, the only rule is that there are four stressed syllables per line (meaning this puppy's written in tetrameter), though they're distributed rather unevenly. Again, this is another thing that makes this poem more akin to a pop song than a more rigorous poetic form, like a sonnet or a villanelle. For example, if you read the poem aloud, you can hear the italicized words stand out more:
Oh where ha'e you been, Lord Randall my son?
Oh where ha'e you been, my handsome young man?
I've been to the wild wood: mother, make my bed soon,
For I'm weary wi' hunting and I fain would lie down.
While these lines aren't all the same length, the stanza achieves a kind of consistency by the fact that the stressed syllables create a solid rhythm that persists throughout the whole poem.
The rhyme scheme of this poem also provides a kind of formal backbone. You'll notice that all of the stanzas have the same four final words: son, man, soon, and down. These are basically paired in two couplets, based on the speakers. Lord Randall's mother's lines always end with "son" and "man," and while these words don't exactly rhyme, they are clearly a matched pair; they're what we call slant rhymes, which means that they look alike, but don't sound perfectly alike. The second couplet, Lord Randall's lines ending in "soon" and "down" actually do rhyme if they're pronounced with the requisite Scottish accent; "down" would originally have been pronounced "doon."
Finally, and perhaps most importantly, the traditional ballad also has some particular stylistic quirks that are unique to it. The one that really stands out here is a little thing called incremental repetition. This phrase sounds oddly scientific, but really, it's pretty simple.
Basically, this means that there is repetition of a set of phrases throughout the song, and each one adds something to the story. A famous example of this formal tool is a song like "The Twelve Days of Christmas" (a lot of Christmas carols are folk songs with their roots in medieval culture, and thus follow the same ballad conventions that "Lord Randall" does). In that song, the cue "On the first (second, third, etc) day of Christmas, my true love gave to me…" is followed by a growing string of bizarre items—a list that grows incrementally, or bit by bit, with each repetition of the formula.
The same thing goes for "Lord Randall," though without the weird list of terrible presents (Seriously: do you really want six geese a'laying? We didn't think so. Seems like a hassle to Shmoop.). The formula here is the set of questions asked by Lord Randall's mother, followed by Lord Randall's response, which is always framed by his refrain. Every time mom asks a question, she uses the same formula, with slight changes; the same goes for her son's answers. As they go on and on with this formal back and forth, we learn more about the story bit by bit, until eventually, at the end, we know what has happened to poor Lord Randall.