Very simply, poetry does not have to rhyme. While there are many more concrete styles of rhyming poetry, poets sometimes feel that non-rhyming poetry can express ideas in ways that rhyming can't. Neither rhyming or non-rhyming poetry is better than the other--it is a matter of personal preference.
Being unconstrained by a rhyme scheme may make it easier to find the right words for your thoughts; however, it is not necessarily easier to write non-rhyming poetry. Quality non-rhyming poetry requires as much effort and skill as good rhyming poetry.
Non-rhyming poetry is a blanket category: within it, there are many more specific types of poetry.
First, we will look at free verse, which is essentially no-rules poetry: no rhyme scheme, no meter, no syllable limits. The poet abides by his or her own boundaries while writing. The best way to get an idea of free verse poetry is to look at a few examples:Earth, My Likeness
by Walt Whitman
EARTH, my likeness,
Though you look so impassive, ample and spheric there,
I now suspect that is not all;
I now suspect there is something fierce in you eligible to burst forth,
For an athlete is enamour'd of me, and I of him,
But toward him there is something fierce and terrible in me eligi-
ble to burst forth,
I dare not tell it in words, not even in these songs.
There is no given meter (or arrangement of the words in the line), and no rhyme scheme, yet Whitman manages to put strong emotional sentiment into his words. When writing poetry, you want to put your thoughts into words in whatever way you think most adequately displays them.
Let's look at one more example of free verse poetry:The Garden
by Ezra Pound
Look at the original version because the formatting is crucial. Notice the tabs he uses--you should not feel forced to line everything up. You are allowed to arrange your poem in any manner you see fit. This can include determining which words are capitalized and which are not--the first word of every line does not always have to be capitalized. The best way to approach free verse may be to just get all of your ideas down on paper and then try to rearrange them to make them flow and give them maximum impact.
Check out these authors for more ideas of free verse (not all of their poems are free verse, though):Ezra PoundWalt WhitmanCarl Sandburg