The tone of a poem is the attitude you feel in it — the writer's attitude toward the subject or audience. The tone in a poem of praise is approval. In a satire, you feel irony. In an antiwar poem, you may feel protest or moral indignation. Tone can be playful, humorous, regretful, anything — and it can change as the poem goes along.
When you speak, your tone of voice suggests your attitude. In fact, it suggests two attitudes: one concerning the people you're addressing (your audience) and one concerning the thing you're talking about (your subject). That's what the term tone means when it's applied to poetry as well. Tone can also mean the general emotional weather of the poem.
Sometimes tone is fairly obvious. You can, for example, find poems that are absolutely furious. The Scots poet Hugh MacDiarmid didn't care for mercenary soldiers (men who fight not because they believe in a cause, but because someone is paying them to fight). Here is MacDiarmid's very angry "Another Epitaph on an Army of Mercenaries":
It is a God-damned lie to say that these
Saved, or knew, anything worth any man's pride.
They were professional murderers and they took
Their blood money and impious risks and died.
In spite of all their kind some elements of worth
With difficulty persist here and there on earth.
Poetry is already so packed with emotion that seeing a poet swearing right at the start may be a shock, but MacDiarmid does exactly that. He makes the disturbing move of insulting the dead soldiers, calling them "professional murderers." Usually, people try not to speak ill of the dead, but evidently MacDiarmid thinks so little of the mercenaries that he feels justified in insulting them. In the last two lines, he implies that, with such evil men in existence, human goodness persists only "with difficulty." These clues lead you to MacDiarmid's tone and his attitude toward his subject: contempt.
Sometimes you can pick up tone from clues in what a person says or writes, as in this untitled poem from the classic Chinese poet Liu Tsung-yüan:
From one thousand mountains the birds' flights are gone;
From ten thousand byways the human track has vanished.
In a single boat, an aged man, straw cloak and hat,
Fishes alone; snow falls, cold in the river.
This poem conveys a tone of melancholy: The birds have abandoned the mountains, and the footprints of human beings (which are signs of human presence) have "vanished" from thousands of roads. The old fisherman you see at the end is all alone, and the word "single," used for his boat, conveys loneliness. The last image is wintry indeed, with snow falling all around him. Taken together, all these elements create an atmosphere of melancholy.
Sharpen your awareness of tone. You'll see it in direct statement, to be sure (as when MacDiarmid cries, "They were professional murderers"), but tone can also reside in:
Images and how they are presented
The implications of a statement or story
The very music and rhythms of a poem