A slant poem is one that uses "slant" or "approximate" rhymes, either internally -- within the line of the poem -- or at the line's end (see reference 1, 2013). An example would be a love poem such as "I love her and I'll make her mine/but I can't make her poem rhyme." "Mine" and "rhyme" are approximate or slant rhymes; they are "close but no cigar" in accurate or full rhyming.
Creating True Rhymes
Writing a slant poem takes several steps for newcomers to poetry-writing, and even well-established poets need these word exercises from time to time. The first step is to create as many full rhymes as one can think of, written only as single words together: "flat/mat," "God/odd," "boat/float" and so on; would-be writers should be able to create at least a dozen or so of these.
The second step is to create simple poems, either in single internally-rhymed lines or couplets, using the created pairings: "I feel that God/Must be quite odd." There should be a poem created for each pairing. This allows amateur poets to become comfortable with full rhyming, and gives experts an excellent "free-writing" experience.
Creating Slants
The third step is to substitute an approximate rhyme for a full rhyme, using the already-completed pairings. This can be done either with consonant repetition -- repeating end letters of words -- or alliteration -- repeating first letters: "God/bad," "flat/flute" "boat/caught."
Finally, the poet takes these new creations and reworks them as couplets or even entire poems, combining them with true rhymes for the best effect: "If God is God/then God is odd; if God is God/then God is bad. Take Him even/take Him Odd; If I'm not God/then I'm not bad." These are the "how" steps to writing slant poetry; the poet should also consider the "why" of slanting.
Who Uses Slants
The best source to answer "why" is the most famous poet to consistently use slant poetry, Emily Dickinson (see reference 2), whose poetry was rejected outright upon its initial publication precisely because her rhymes were not exact. In fact, numerous editors sought to "correct" Dickinson by rewriting her work to manufacture true rhymes. These corrupted versions are gone, and the poet's original poetic forms are available everywhere.
Dickinson used slants for several reasons: she wanted to startle the reader's sensibilities, she wanted to encourage one to see unlike things in comparison and on occasion she simply seems to have felt like it. Most frequently, however, her slants aim at personifying an abstraction with a concrete image, as in the work she simply called "Poem 1260," (see reference 3, 2004), aka "Because that you are going," where she rhymes "he besides concedes," a concrete action, with "confiscated Gods." The line in context presents God as agreeing to the human worship of other things, an unreality rendered brilliantly real.
Why Use Slants
A poet uses slants for the same reasons Dickinson did: to shake up his/her readers from their neatly rhymed preconceptions, to juxtapose dissimilar things and actions into a poetic whole, and also because -- well, it simply feels right. If poets are writing slant poetry to introduce abstractions as concrete ideas, to show another side of an issue or to reveal a truth, Dickinson favored that, too; she herself advised all writers to "tell all the truth but tell it slant."