Definition of Assonance:
Repetition of vowel sounds without repetition of consonants.  Often used as an alternative to rhyme in verse.

Example of assonance #1:


 And the Raven, never flitting, still is sitting, still is sitting
On the pallid bust of Pallas just above my chamber door;
    And his eyes have all the seeming of a demon’s that is dreaming,
    And the lamp-light o’er him streaming throws his shadow on the floor;
And my soul from out that shadow that lies floating on the floor
        Shall be lifted—nevermore!

"The Raven" is a well known poem and is a great example of assonance.  "Seeming of a demon's" repeats the "ee" sound.  Edgar Allan Poe in this case uses assonance and internal rhyme in the same stanza.

Example of Assonance #2:

Behind Me Dips Eternity, by Emily Dickinson

Behind Me -- dips Eternity --
Before Me -- Immortality --
Myself -- the Term between --
Death but the Drift of Eastern Gray,
Dissolving into Dawn away,
Before the West begin --

This poem also uses assonance through repetition of the "ee" sound, as in "Behind Me...".   This poem uses both assonance and consonance.

Example of Assonance #3:


I hear America singing, the varied carols I hear;
Those of mechanics - each one singing his, as it should be, blithe and strong;
The carpenter singing his, as he measures his plank or beam,
The mason singing his, as he makes ready for work, or leaves off work;

The use of assonance is more subtle in this poem.  The repetition of the soft "e" sound in "America" & "Varied" creates the effect of assonance. 

Assonance in poetry.

Slideshow:    Assonance

Assonance Video

See also:   Alliteration    Consonance