Colorless green ideas sleep furiously
"Colorless green ideas sleep furiously" is a sentence composed by Noam Chomsky in 1957 as an example of a sentence whose grammar is correct but whose meaning is nonsensical. It was used to show inadequacy of the then-popular probabilistic models of grammar, and the need for more structured models.
The full passage says:
- Colorless green ideas sleep furiously.
- Furiously sleep ideas green colorless.
It is fair to assume that neither sentence (1) nor (2) (nor indeed any part of these sentences) had ever occurred in an English discourse. Hence, in any statistical model for grammaticalness, these sentences will be ruled out on identical grounds as equally "remote" from English. Yet (1), though nonsensical, is grammatical, while (2) is not.

# Meaninglessness
While the meaninglessness of the sentence is often considered fundamental to Chomsky's point, Chomsky was relying upon this only to ensure that the sentences had never been spoken before.  Thus, even if one were to prescribe a likely and reasonable meaning to the sentence, the grammaticalness of the sentences are concrete despite being the first time a person had ever heard that phrase, or those words in such a combination.  This is also a counter-example to a challenging idea at the time that the human speech engine was based upon a Markov Chain, and simple statistics of words following others.
The sentence can be given an interpretation through polysemy. Both green and colorless have figurative meanings, which still make us able to interpret colorless as "nondescript" and green as "immature" or "environmentally-friendly". So the sentence can be constructed as "nondescript immature ideas have violent nightmares", a phrase not unimaginable in poetry. In particular, the phrase can have legitimate meaning too, if green is understood to mean "newly-formed" and sleep can be used to figuratively express mental or verbal dormancy. An equivalent sentence would be "Newly formed bland ideas are inexpressible in an infuriating way." One meaning could be "unimaginative environmentalist ideas are unpopular".
Writers have attempted to provide the sentence meaning through context, the first of which was written by Chinese linguist Yuen Ren Chao. A literary competition was held at Stanford University in 1985, in which the contestants were invited to make Chomsky's sentence meaningful using not more than 100 words of prose or 14 lines of verse.
An example entry from the competition, from C.M. Street, is:

# Other examples
There is at least one earlier example of such a sentence, and probably many more. The pioneering French syntactician Lucien Tesniere came up with the French sentence "Le silence vertebral indispose la voile licite" ("The vertebral silence indisposes the licit sail").
The game of cadavre exquis (1925) is a method for generating nonsense sentences.  It was named after the first sentence generated, Le cadavre exquis boira le vin nouveau (the exquisite corpse will drink the new wine).
There are doubtlessly earlier examples of such sentences, possibly from the philosophy of language literature, but not necessarily uncontroversial ones, given that the focus has been mostly on borderline cases. For example, followers of logical positivism held that "metaphysical" (i.e. not empirically verifiable) statements are simply meaningless; e.g. Rudolph Carnap wrote an article where he quite literally claimed that almost every sentence from Heidegger was grammatically correct, yet meaningless. Of course, some philosophers who were not logical positivists disagreed with this; at the same time, many who had tried to read Heidegger agreed completely.
Another example is Groucho Marx's quote, "Time flies like an arrow. Fruit flies like a banana" which mixes syntactic confusion with semantic confusion and thus indicates the problems (again) of a purely syntactical approach to parsing natural language without semantic context.
The philosopher Bertrand Russell used the sentence "Quadruplicity drinks procrastination" to make a similar point; W.V. Quine took issue with him on the grounds that for a sentence to be false is nothing more than for it not to be true; and since quadruplicity doesn't drink anything, the sentence is simply false, not meaningless.
Examples like Tesniere's and Chomsky's are the least controversially nonsensical, and Chomsky's example remains by far the most famous.
John Hollander wrote a poem titled "Coiled Alizarine" in his book, The Night Mirror. It ends with Chomsky's sentence.
Clive James wrote a poem titled "A Line and a Theme from Noam Chomsky" in his book, Other Passports: Poems 1958-1985. It opens with Chomsky's second meaningless sentence and discusses the Vietnam War.
Van Dyke Parks, in the lyrical masterpiece Surf's Up first performed by The Beach Boys and later Brian Wilson, provides an excellent example in the line 'Columnated ruins domino'. When one looks at the word 'domino' as a verb, it is possible to picture ruined columns falling like dominoes, but the word structure is nonsensical upon first glance.
Stephen Fry delivers the following line in an A Bit of Fry and Laurie sketch entitled Language Conversation: "I can say this sentence and be confident it has never been uttered before in the history of human communication: "Hold the newsreader's nose squarely, waiter, or friendly milk will countermand my trousers.""
Another approach is to create a syntactically-correct, easily parseable sentence using nonsense words; a famous such example is "The gostak distims the doshes".  Lewis Carroll's Jabberwocky is also famous for using this technique, although in this case for literary purposes.
Other "meaningless utterances" are ones that make sense, are grammatical, but have no reference to the real world, such as "The present Queen of France rides a unicorn."  There is no such person as the present Queen of France (France is a Republic and has been so for some time; a monarchist might say that the Duchess of Orleans is the rightful Queen of France, but there is no actual Queen) and there are no such things as unicorns.

# Reactions towards the notion of meaninglessness
Many functionalist linguists and cognitive linguists, most notably Dwight Bolinger, George Lakoff, Thomas Givon, William A. Croft and M.A.K. Halliday, have argued against the notion of meaninglessness in language, arguing that the purpose of language is communication; that is, the exchange of meanings. One of their arguments is that, while sentences like 'colorless green ideas sleep furiously' may be possible, they hardly ever appear in naturally occurring language.

# Challenges
Fernando Pereira of the University of Pennsylvania has fitted a simple statistical model to a corpus of newspaper text, and shown that under this model, "Furiously sleep green ideas colorless" is about 200,000 times less probable than "Colorless green ideas sleep furiously".
This statistical model defines a similarity metric, whereby sentences which are more like those within a corpus in certain respects are assigned higher values than sentences less alike. Pereira's model does assign an ungrammatical version of the same sentence a lower probability than the syntactically correct form. However, it is not clear that the model assigns every ungrammatical sentence a lower probability than every grammatical sentence. That is, "colorless green ideas sleep furiously" may still be statistically more "remote" from English than some ungrammatical sentences.