The colloquial see you later, alligator, which originated in American English, is a catchphrase used on parting. The expected response is in, or after, a while, crocodile.
—Cf. also notes on ‘see you later, agitator’.

The earliest instance of see you later, alligator that I have found is from Teenagers’ Slang Expressions Are Explained by Columnists, by “Jackie and Jane, Star-Bulletin Teen Columnists”, published in the Honolulu Star-Bulletin (Honolulu, Hawaii) of 1^st May 1952:

Lots of the Hawaii-style slang can be credited to or blamed on the Mainland. Some of it’s strictly jive talk. And of course there’s the additional “pidgin” and Hawaiian words to dress it up.
[…]
Besides the everyday “slang,” many high school students use expressions such as “toodle-oo tofu,” “so long, dai-kong,” or “see you later, alligator.”
These sayings invite expressions like “see you soon, goon,” and “hit the road, toad.”

On 15^th February 1954, The Kansas City Times (Kansas City, Missouri) published Words, Wit and Wisdom, in which William Morris wrote:

It has long been my conviction that the most effervescent and everchanging department of the American language is the subdivision labeled “Teenage slang.” So I suggested that my younger readers send in the favorite expressions of their teenage set.
Well the mails have brought literally hundreds of letters and postcards from youngsters all over the country. Here are just as few and catchphrases like:
[…]
Melt down and float away: meaning get lost, drop dead, or in grown-up language, go away.
Go dad: meaning wonderful.
Solid Jackson: meaning excellent.
See you later, alligator: meaning good-bye.

On 28^th February 1954, The Sunday Sun (Baltimore, Maryland) published Do Kids Speak English?, in which Lester Rand, president of the Youth Research Institute, “an organization which exclusively surveys the tastes, attitudes and buying habits of young people five through 25”, explained that youngsters develop their own talk as a way of excluding grownups; about see you later, alligator, he said:

“The ‘alligator’ is an all-encompassing term and relieves the party of having to recite several names.”

The author of the article, Beulah Racklin, wrote that, additionally:

Rhyming expressions, which are mainly for effect rather than to convey any actual meaning, are very popular and somewhat confusing like ‘Do you know what I mean, jellybean?’ ‘Let me have steak, Jake,’ ‘Have a piece of salami, Tommy,’ etc. In few instances does the person addressed comply with the speaker’s demands. In most cases names are changed for ones that rhyme.

So popular was the phrase that in 1955 it was one of the twenty-four slogans printed on the badges offered by Armour Star Franks, a brand of frankfurters; the following advertisement appeared in the Akron Beacon Journal (Akron, Ohio) on 14^th August of that year:

Read ’em and laugh! Wear ’em and be a leader in your crowd! These Breezy Buttons that come in packages of Armour Star Franks are real c-o-o-l. They show you are hep to the latest jive talk. There are 24 different breezy buttons. Each one with a different slogan printed on it — plus a real laugh getting cartoon or smart design! And they are valuable, too. Made of strong metal with stay-on pins that’ll hold ’em tight to your cap, shirt or belt. And that isn’t all, kids! Each button comes in two big bright colors like lemon and red, pink and green, orange and blue — to name a few! Gee! There are 26 different color combinations that you can trade! You’ll want to get the most colors besides a complete set of breezy slogans!

two illustrations for the advertisement for Armour Star Franks
from the Akron Beacon Journal (Akron, Ohio) – 14^th August 1955

 

See You Later, Alligator, a song written and first recorded in 1955 by the American singer-songwriter Bobby Charles (Robert Charles Guidry – 1938-2010), capitalised on the popularity of the catchphrase among children and teenagers.

(In the Catalog [sic] of Copyright Entries (Washington, 1956), it appeared as See Ya Later, Alligator, words and music by Robert Charles Guidry, copyright claimed by Robert Charles Guidry, registered on 6^th September 1955, as number EU409054.)

The most popular recording of the song, however, is that made by Bill Haley* & His Comets later in 1955; it appears for example in this advertisement for Millikan’s Records, published in The Hammond Times (Hammond–East Chicago, Indiana) on 30^th December 1955:

(* William John Clifton Haley (1925-81))

 

Lyrics of See You Later, Alligator (1955), by Robert Charles Guidry:

Well, I saw my baby walkin’ with another man today
Well, I saw my baby walkin’ with another man today
When I asked her what’s the matter
This is what I heard her say

See you later alligator, after ’while crocodile
See you later alligator, after ’while crocodile
Can’t you see you’re in my way now
Don’t you know you cramp my style

When I thought of what she told me, it nearly made me lose my head
When I thought of what she told me, it nearly made me lose my head
But the next time that I saw her
I reminded her of what she said

See you later alligator, after ’while crocodile
See you later alligator, after ’while crocodile
Can’t you see you’re in my way now
Don’t you know you cramp my style

She said I’m sorry pretty daddy, you know my love is just for you
She said I’m sorry pretty daddy, you know my love is just for you
Won’t you say that you’ll forgive me
And say you just love me too

I said wait a minute ’gator, I know you meant it just for play
I said wait a minute ’gator, I know you meant it just for play
Don’t you know you really hurt me
And this is what I have to say

See you later alligator, after ’while crocodile
See you later alligator, after ’while crocodile
Can’t you see you’re in my way now
Don’t you know you cramp my style