A rhyming couplet is a set of lines, back to back, that rhyme. Usually they have the same meter so that they appear as a coherent whole.
Example: Geoffrey Chaucer's The Canterbury Tales
is written in rhyming couplets. (Beware, though: it is in Middle English).
From the prologue:
Singing he was, or fluting all the day;
He was as fresh as is the month of May.
This couplet is in iambic pentameter.
A rhyming quatrain is a set of four lines that follow a rhyme scheme. The rhyme schemes could be:
AABB--(the first and second lines rhyme with each other and the third and fourth lines rhyme with each other. In other words, a pair of rhyming couplets together)
ABAB--(the first and third lines rhyme with each other and the second and fourth lines rhyme with each other)
ABBA--(the first and fourth lines rhyme with each other and the second and third rhyme with each other)
ABCB--(the second and fourth rhyme with each other).
Again, quatrains usually have a given meter so that they sound like they flow together.
Some examples: The Hippopotamus
by Ogden Nash
(my favorite poet of all time)
Behold the hippopotamus!
We laugh at how he looks to us,
And yet in moments dank and grim,
I wonder how we look to him.
Peace, peace, thou hippopotamus!
We really look all right to us,
As you no doubt delight the eye
Of other hippopotami.
Note the AABB rhyme scheme.
Here's Look Back on Time with Kindly Eyes
by Emily Dickinson
Look back on time with kindly eyes,
He doubtless did his best;
How softly sinks his trembling sun
In human nature's west!
This is an ABCB rhyme scheme.
The couplet and the quatrain can be used as building blocks for your poems, as we will see next.