This is a difficult question. Whitman's short, rhyming poem "O Captain! My Captain" is probably his most widely read piece in high school classrooms, but Song of Myself is almost certainly his most widely read collection in the college classroom.
"When Lilacs in My Dooryard Last Bloomed" may be Whitman's greatest achievement in a single long poem, however, as it encapsulates many of his passions, themes, and poetic abilities around a loosely focused topic - the hero's death.
Whitman ushered in a whole new era of poetry!
Out with rhyme scheme, out with meter, out with structured stanzas!
The poets before him were heavily influenced by British poetry, but good ol' Walt changed all that! Emily Dickinson appeared at around the same time and she, too, changed the face of poetry.
Also, Leaves of Grass has sexual content to it (and some will say homosexual content--I have not researched enough to know if he was gay or not, but have heard he was), which was not present in as much detail prior to this.
You'll probably get a lot of different answers to this question, but I'll start. Emerson called for a "new" American literature, a literature worthy of the scope, size, and newness of the American country. Whitman responded to this request. He threw aside the conventions of earlier poetry and gave us a poetry worth of our democracy, a poetry that attempted to encompass the immense variety of people/nationalities/races that contributed to the American of Whitman's time. Whitman also brought an almost mystical vision to his writing, seeing everything as a part of a great whole, everything somehow joined to everything else, and, in his almost endless lists, where things just appear in no particular order, everything equal to everything else.
Emerson recognized Whitman's greatness, and in a famous letter celebrated him: "I greet you at the beginning of a great career, which yet must have had a long foreground somewhere, for such a start. I rubbed my eyes a little, to see if this sunbeam were no illusion; but the solid sense of the book is a sober certainty. It has the best merits, namely, of fortifying and encouraging."
A good place to start studying Whitman is his "Democratic Vistas" essay. E-notes provides a good summary at the source below.
Hope you enjoy your study of Whitman!
It is hard to really put a finger on the “one” thing that was Whitman’s greatest contribution to American literature, but I would have to say his influence on poetry stands out. Walt Whitman is known as the father of free verse, which is a style of poetry that does not conform to the traditional rhyme and meter of earlier forms. His rather bohemian lifestyle was also an inspiration to the poets of the Beat movement, such as Allen Ginsberg and Jack Kerouac. With the publication of Leaves of Grass, he certainly became America’s poet. For many, the “I” at the center of “Song of Myself” is a representation of the American people.