A quatrain is four lines of poetry that rhyme.  It is the most common type of poetry in English. 
In some quatrains, all four lines rhyme--ABAB.
In others, only every other line rhymes--xByB. 
There are other variation, such as the second quatrain of "the Good Morrow" below, which is AAxA. 

The Good-Morrow
I wonder, by my troth, what thou and I
Did, till we loved?  Were we not weaned till then,
But sucked on country pleasures, childishly?
Or snorted we in the seven sleeper's den?

`Twas so; `cept this, all pleasures trifles be.
If ever any beauty I did see,
Which I desired, and got,
`twas but a dream of thee. 

   John Donne

And, as in uffish thought he stood,
The Jabberwock, with eyes of flame,
Came whiffling through the tulgey wood,
And burbled as it came!

Lewis Carrol, the Jabberwock

This is my Father's World

This is my Father's world
And to my listening ears
All Nature sings and `round me rings
The music of the spheres. 

Maltbie D. Babcock

Tyger! Tyger! burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?

From William Blake's "The Tyger"