While contemporary critics tend to judge poems by the line, and sometimes by the phrase, this leaves out most of the work in poems longer than a dozen lines are so. Powerful longer poems work not only because of fireworks in a given line, whether a strong individual metaphor or a startling image, but because each has an underlying structure that provides the foundation for whatever stanza, line, meter and foot, figurative language, logic, and narrative development are doing with the content. An idea, the arc of a story, a character's development in a dramatic monologue, such are not stated but are the scaffolding that everything else hangs on. Just as you don't see a copy of a screenplay's descriptions and directions while watching a movie, neither do you hear an explicit intention or structural direction in a poem. But if there isn't one, the poem will fail without an accompanying explanation by the poet, invariably a sign of an unfinished piece.
Structure is not the "message" of a piece. There are messages to the queen's censor in the players scene in Hamlet (stop messing with our theaters!) but the dramaturgy of the play is not very different from that used by any other Elizabethan playwright (or by most contemporary filmmakers).
Structure is not the theme of a piece. The theme of a piece is a mixture of what a reader gets out of a poem and what the author intended to put in, both of which are notoriously difficult to measure. If you state the structure, you won't find the theme.
Structure is not the story of a piece. The "argument" preludes to sections of long narratives, such as Paradise Lost, don't describe how the section or poem works, but give a plot summary.
In a short piece, such as a sonnet, the structure is simple, for example argument/counterargument.
faith I do not love thee with mine eyes,
For they in the a thousand errors note;
But 'tis my heart that loves what they despise,
Who, in despite of view is pleased to dote.
Nor are mine ears with thy tongue's tune delighted;
To tender feeling to base senses prone.
Nor taste nor smell desire to be invited
To any sensual feast with thee alone.
Okay, we get the point. After the first four lines, the rest
detailing, important, but detailing nonetheless. This is the
argument of sonnet 141. Then comes the counter-
argument, in the same voice, but the side of the story
as yet untold.
my five wits nor my five senses can
Dissuade one foolish heart from loving thee,
Who leaves unsway'd the likeness of a man,
Thy proud heart's slave and vassal wretch to be.
Only my plague thus far I count my gain,
That she that makes me sin awards me pain.
Okay, he can't resist someone he finds repulsive; that's obvious on the surface. He spends eight lines describing why he doesn't love her. That's the argument, though this is not a formally acknowledged term. It's just useful. Then he spends the last six lines describing why he can't resist her. That's the counterargument; the jarring transition, from a discussion of love in the first eight lines to a plain discussion of want in the last -- the two are frequently unrelated (no news), is bound together by the concluding couplet, where the pleasure of sex is reduced to sin, however ironically, and the woman to an ugly temptress. The structure of the sonnet, while nowhere explicit, makes this transition easy. If the subject matter of the poem were not suitable to that structure, it would require a different form.
In a short narrative, it may be action/consequence. In Robert Frost's "The Black Cottage," from North of Boston, a poem of about 100 lines, the action is the minister's willingness to tell about his memories of the late owner of the cottage in question. The consequence is the revelation of the minister's hypocrisy. Frost's use of the structure suits the subject matter; it could be used and has been in tens of thousands of other kinds of short narrative poems and stories.
In a play, it may be situation/conflict/action/consequence/resolution and there may be one more than one such through-line in a play, or more than one action and consequence, provided they're somewhat related to one another. At the beginning of the comedy Who's Afraid of Virginia Woolf, Martha is attracted to a young math professor, and her husband George is out to stop it. He plays Get the Guests to humiliate the young couple. The consequence is that Martha is so offended that she makes love with the math professor as much out of spite as attraction and George is humiliated. There is a subsequent action by George to get even, where he leads Martha to the revelation that the son they claim to have is a fantasy, which shatters Martha's illusions. The resolution is that they find a sort of peace together. Albee employed traditional dramatic structure; much the same underlies tragedies by Sophocles or tragicomedies by Shaw.
In a lyric the structure may be detail/generality or vice versa. From Frederick Feirstein's "Hurricane"...
The eye becomes Manet's or Debussy's,
On this inch thick lawn facing the beach
Where blue and white casabellas match
The white-capped-green-to-nearly-cobalt sea
Held momentarily by swayback palms,
Miniature hills of purple, heart-shaped flowers...
The details, which carry the poem to its end....
The best lyric poems all use this simple structure; the work of course
is filling the structure out. But without it, there's little point
Combinations of these are common in poetic drama and narrative. They're never really visible, only felt, as when the tide comes in and then goes out. When the tide comes is about one thing; when the tide goes out is about another. Only the poet decides that.