Since this poem is not divided into stanzas, I'll separate into sections which make sense for meaning.

I dream a world where man
No other man will scorn,

The speaker opens with his vision of a new world, one where each person refrains from judging other people as worthless. This start to the poem immediately places individual accountability on each person. The speaker isn't just dreaming of a world where no one receives scorn, but instead on a world where no one does the scorning. This makes a difference, turning people into active participants instead of passive recipients.

Where love will bless the earth
And peace its paths adorn

If humankind avoided condemning others, the very tone of interactions would change. Love would fill the earth, leading to peaceful interactions in brief encounters and in meaningful relationships.

I dream a world where all
Will know sweet freedom's way,

The speaker dreams of a world where everyone knows freedom. The world is free of slavery, of kidnappings, and of indentured servitude. In this world, people are free to think and live in accordance with the desires of their hearts.

Where greed no longer saps the soul
Nor avarice blights our day.

In this quest, people no longer attempt to fill their souls through desires based in greed. Thus, the focus of each day is not simply to acquire more wealth.

A world I dream where black or white,
Whatever race you be,
Will share the bounties of the earth
And every man is free,

In this world of the speaker's dreams, all races share the earth's resources and riches equally. Because of this willingness to live in harmony with the earth and each other, it is possible for each person on the earth to live in freedom.

Where wretchedness will hang its head
And joy, like a pearl,
Attends the needs of all mankind-
Of such I dream, my world!

This new Earth would be the end of misery, personified here by its hanging head. Joy is compared to a pearl, one of those rich treasures of the earth mentioned earlier, a priceless treasure that all of mankind can experience. The speaker concludes by claiming the world and all of its inhabitants as his own ("my world"), propelling his hope of this dream and longing for its realization.