The poem known to most is actually just the first verse of a longer poem; much like The Star Spangled Banner, most people only know the first verse, and most of those don't realize there are more.

The poem, titled "The Star", first saw publication in 1806. It was written by Ann and Jane Taylor (sources differ on whether one or the other wrote it, or if they wrote it together; most say it was just Jane) for their book, "Rhymes of the Nursery."

The music it is set to is a piece first published by M. Bouin, "Ah! Vous Dirai-Je Maman", in "Les Amusements d'une Heure et Demy" in 1761. (Another source claims the title was "Les amours de Silvandre"). Many composers wrote variations on this theme, most notably Mozart, who used them as keyboard exercises. (K265/300e) The original author, however, remains unknown.

The poem and the music probably came together in 1838, in "The Singing Master: First Class Tune-Book"; another source places the date as 1881, in "The Franklin Square Song collection".

Twinkle, twinkle, little star,
How I wonder what you are!
Up above the world so high,
Like a diamond in the sky.

When the blazing sun is gone,
When he nothing shines upon,
Then you show your little light,
Twinkle, twinkle, all the night.

Then the trav'ller in the dark,
Thanks you for your tiny spark,
He could not see which way to go,
If you did not twinkle so.

In the dark blue sky you keep,
And often thro' my curtains peep,
For you never shut your eye,
Till the sun is in the sky.

'Tis your bright and tiny spark,
Lights the trav'ller in the dark:
Tho'I know not what you are,
Twinkle, twinkle, little star.

An interesting aside is that a recent (1998) children's book based loosely on the poem claims a Jane Taylor as illustrator. It seems safe to say that this is not the same Taylor, but it's a nice coincidence.
The famous first verse has been translated into various languages, with varying results regarding meter and melody. For example, this version in obfuscated English (author unknown):

Scintillate scintillate globule vivific
fain would I fathom thy nature specific
loftily poised in the ether capacious
strongly resembling a gem carbonaceous.

while keeping to the spirit of the poem, cannot easily be sung to the associated tune.

Gardner, Martin and Carroll, Lewis. The Annotated Alice (The Definitive Edition). W. W. Norton, 2000