The words "modern" and "modernist" are both disputed descriptions of poetry and several other arts. Matters for dispute include the period covered and the characteristics of modernity and modernism. G. K. Chesterton, who wrote traditional poetry at the time when such modernists as T. S. Eliot and Ezra Pound were conducting their experiments, famously complained that "modern" was an absurdly weak and unphilosophical description to attach to a way of writing. In "The Case for the Ephemeral," he wrote:
It is incomprehensible to me that any thinker can calmly call himself a modernist; he might as well call himself a Thursdayite.
Chesterton's own verse, however, makes the opposite point. He is not a modernist, even though his poems were written during the modernist period. There must, therefore, be some definite characteristics which identify modernism as a school and a style, and these can be found by comparing Chesterton and other traditional poets to Eliot and the modernists.
First, the modernists tended to use free verse. This is not invariable, and they certainly did not dispense with rhyme, meter, and form altogether. However, they were willing to vary the form to suit the subject, as Eliot does in "The Love Song of J. Alfred Prufrock." Second, there is a complexity and doubtfulness about modern poetry which is not found in more traditional verse. Take "The Love Song of J. Alfred Prufrock" again. This is not a simple poem about unrequited love, like an Elizabethan lyric in which the poet assumes that all would be well if his mistress returned his feelings. Prufrock questions everything and can see no possibility even of revealing his love—an act which seems to him to be a cataclysm which would "disturb the universe."
Beyond these two characteristics, modern poetry was reflective of the age in which it was written in a way that no previous school or period had been. One can argue that Kipling and Tennyson incidentally tell the reader quite a lot about what it means to be a Victorian, but this is not their subject, precisely because their Victorian values remain unquestioned. The equivocal, doubtful tone characteristic of modern poetry means that the reflection of the age is usually an uncomfortable one, as when Eliot, in "The Waste Land," continually contrasts the heroism and splendor of past ages with the squalor of London in the twentieth century.
In addition to the points made by the first two educators, it should also be noted what Modernism was essentially about, the reason it emerged as it did. After the formal, flowery, and romantic era of Victorian literature, Modernist works leaned towards experimentation. Modern poets wished to both extract inspiration from diverse works of the past and ground poetry such that its language and meaning was more accessible to the average person.
With the end of the nineteenth century and two World Wars, the Modernists wished to comment candidly and competently on the degenerating state of the world. This context helps to explain the pervasive characteristics of Modern poetry: free verse and otherwise untraditional forms, disillusionment and a preoccupation with perception, and how to cope with a fragmented reality.
A more in-depth look at the Modernists can be found here: https://www.enotes.com/topics/modernists.
Overall, Wallace Stevens captured the essence of the period in his poem "Of Modern Poetry," when he wrote:
It has to be living, to learn the speech of the place.It has to face the men of the time and to meetThe women of the time. It has to think about warAnd it has to find what will suffice.
Modern poetry often features disrupted syntax, which refers to irregular sentence structures. In addition, many modern poems feature a stream of consciousness presentation in which the narrator presents the thoughts that come to his or her mind without regard to sequence or logic. Stream of consciousness mirrors the way in which the subconscious mind works and shows poets' increasing interest in psychology. An example in T. S. Eliot's "The Love Song of J. Alfred Prufrock" is the following:
"I grow old ... I grow old .../ I shall wear the bottoms of my trousers rolled. /Shall I part my hair behind? Do I dare to eat a peach?"
In this poem, the narrator presents his thoughts in a jumbled fashion, similar to the way in which thoughts pop into our conscious mind out of our subconscious mind. Thoughts about momentous subjects, such as aging, are combined with fleeting thoughts about whether the narrator should roll his pants or eat a peach.
Modern poets also convey a sense of alienation from the world. As Eliot writes in "Prufrock," "I have heard the mermaids singing, each to each. /I do not think that they will sing to me." The poet does not believe he can experience the wonders or delights of the world; instead, he is alienated and distanced from its experiences and marvels.
The single most common characteristic of modern poetry (in the European and American traditions, at least) is probably open form and free verse, which is quite different from the fixed forms and meters of traditional poetry. A reader of high-brow poetry today sometimes has to look around a bit to find modern sonnets or even ballads or other poems with regular line length, stanza length, meter, and end rhyme.
A second characteristic might be called fragmentation, juxtaposition, intertextuality (reference to other poems or other writings), and allusion. For an example of all of the above, see T.S. Eliot's long poem The Waste Land.
Not all recent poetry is "modern," of course. If this is an assignment, you may want to consider putting two poems from different centuries side by side -- two love poems, one by William Shakespeare and another by e.e. cummings -- and seeing what sorts of differences emerge.