An easy comparison is to compare "Poetry" to that of the structure of a standard sonnet. By definition a sonnet is a 14 line poem. Moore's poem is 31 lines. So, it's not a sonnet.
A Petrarchan sonnet will have a specific rhyme scheme over those 14 lines. The first 8 will be abbaabba. The next 6 will typically be cdcdcd. A Shakespearean sonnet will be abab, cdcd, efef, gg over the 14 lines. Moore's poem doesn't follow a rhyme scheme, so it doesn't compare nicely to a sonnet in that regard either.
Any English sonnet is also almost always written in iambic pentameter. Moore's poem doesn't follow a set rhythm. It is written in free verse. However with that said there are specific syllable counts. Line one of each stanza is always 19 syllables. More often than not, line 2 is 21-22 syllables. Again, more often than not, the third line of each stanza is right around 10-12 syllables.
The other cool thing to note about her poem is that because of her goofy line breaks (enjambment), each next line continues toward the right of the poem a little bit further than the previously enjambed line. Looks like a stair case of sorts. And it gives the effect that her thoughts and words are gaining a sense of momentum.
Comparing "Poetry" to a traditional ballad doesn't get much better. A traditional ballad typically tells a story. It has a main character and plot. Moore's poem doesn't do that. It's a telling of the speaker's thoughts and emotions about poetry. Ballads are also broken into 4 line stanza with abab rhyme scheme. As I said before, Moore's poem does not rhyme. But you might be able to make a claim that each stanza is grouped into 4's. The enjambment happens four times per stanza.