Katherine Pierpoint's "Swim Right Up to Me" is a narrative poem, which means it tells a story. In this case, the narrator (which I presume is a female but may not be) is looking back in time and remembering how she used to "swim" on the tapestried piano stool which she used to place in the middle of her father's study.
A narrative poem can rhyme or not, and this one does not. A poem which does not have a particular rhyme or rhythm is called free verse. The eNotes Guide to Literary Terms (linked below), describes free verse poetry beautifully as this:
verse that lacks regular meter and line length but relies upon natural rhythms. It is free from fixed metrical patterns, but does reveal the cadences that result from alternating stressed and unstressed syllables. The form is thought to add force to thought and expression. While giving...
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an address on May 17, 1935, Robert Frost explained, “Writing free verse is like playing tennis with the net down.”
Notice that Pierpoint's poem contains lines of varying length and has no discernible rhyming pattern. While it does not have any of those things, it does have the flow and rhythm of natural speech--as if the speaker were simply telling a story, which she is.
A narrative poem, like a story, has to have a beginning, middle, and end. In this case, the narrator starts by putting the stool in the center of the room, and then she swims.
Stomach down, head up, arms and legs rowing hard;
I swam bravely, ploughing up the small room,
Pinned on a crushed stuckness of stomach to tapestry,
The twin handles hard on my elbows on the back stroke.
Soon (in the middle) her mother is shouting encouragement at her, reminding her to breathe, and wiping the sweat from her hair and eyes. Finally (the end), the speaker decides she has gotten good enough at swimming that she plans to learn how to fly, next. For her it will be easy, and she will fly
Higher than the kitchen table, even. The garden wall.
The tone of the poem is nostalgic, and the story is told with plenty of remembered imagery, such as the feel of the stool beneath her, the sweat on her brow, her mother's encouragements, and being
Pinned on a crushed stuckness of stomach to tapestry.
Frankly, it is much easier in one sense to discuss the form of a perfectly metered poem because there are specific elements to discuss. In a free verse narrative, there is no particular rhyme, rhythm, or format to identify; instead we simply "hear" the poem as the narrator "speaks" the memory. It feels natural and we are moved by the details of the remembrance. "Form" in this poem, then, is more about how a memory is evoked than about its structure. Pierpoint uses sensory images to evoke the sense of "swimming" in the middle of an office, and when the speaker tells her story, we are reminded of our own memories of learning and doing such things.