A rhyme that is inexact, imperfect, or altered. This may be accomplished through slight variation from identical vowel sounds, consonants, syllabic structure, or points of emphasis in each word. Slant rhymes are used for a variety of specific poetic purposes. I would venture that the most common purpose is to draw attention to a key part of a poem (or other rhyming work, I suppose).
Slant rhymes can be used anywhere one might use rhyme. Traditionally, rhyming poems keep their rhymes at the end of their lines. However, as in the previous sentence ("rhyme" and "line", yo--while they contain the same syllabic structure and vowel sound, one has an "n" and the other an "m"), one may use rhyme anywhere it serves a good purpose. So, slant rhymes, as a type of rhyme, can be used anywhere one might use straight rhyme, but usually for a different purpose.
Poems can accomodate a startling variety of aural gymnastics. Internal rhyme, feminine or masculine rhyme, and repetition all have their uses--they give a poem a rhythmic structure, put emphases on different syllables, create interest in general. A regular rhyme or repetition is a fairly standard way to accomplish any of these goals. Slant rhyme, however, stands as a definite contrast to straight rhyme, especially if used in close conjunction. It is not used to establish a rhythm, but to break it in a crucial place.
In a poem such as Thomas Hardy's "The Darkling Thrush", for example, the great majority of the rhymes are straight. The third stanza, however, contains several slant rhymes:
At once a voice arose among
The bleak twigs overhead
In a full-hearted evensong
Of joy illimited;
An aged thrush, frail, gaunt, and small,
In blast-beruffled plume,
Had chosen thus to fling his soul
Upon the growing gloom.
The pair "among" and "evensong" have different lengths and emphases as well as slightly different sounds in the "o" vowel. "Overhead" and "illimited" have different emphases on different parts of each word: "overhead" contains accents on the first and last syllables, while "illimited" only has an accent on the second syllable. Overall, however, they are fairly close. From this, I would guess that these pairs are intended to point to the third and most dramatic slant set in the stanza, that of "small" and "soul". These two words, while containing many similar consonant sounds, have entirely different vowel sounds--a short "a" is paired with a long "o".
What is it about this pair, then, that Hardy wanted to emphasize? This is the darkling thrush of the title, certainly, and the key image in the poem, so perhaps he merely wanted to show us how important it is. But I think he also wanted to show the reader why this image is important. It establishes a parallel between the bird's physical appearance and its apparent inner appearance, in the form of its voice. From this, we can deduce that although the thrush is singing in a gallant and wholehearted manner, perhaps everything is not as it should be; perhaps the bird's soul is closer to its fallible physical body than it (or Hardy) would ideally want to believe.
Here, slant rhyme establishes a feeling of general wrongness within an already depressing poem. It makes the reader feel that it's not just a blustery day out; something is actually wrong not only with the bird, with the poem's speaker, or with the structure of the poem, but with the entire structure of the universe.