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What is recursion? I've looked at the Wikipedia's explanation (recursion and then recursion in language) but that explanation is not really clear.

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Recursion is a property of language.

From a Linguistics viewpoint, recursion can also be called nesting. As I've stated in this answer to what defines a language (third-last bullet point), recursion "is a phenomenon where a linguistic rule can be applied to the result of the application of the same rule."

Let's see an example of this. Consider the sentence:

Alex has a red car.

An application of recursion would give:

Alex, whom you know very well, has a red car.

And then:

Alex, whom you know very well, has a red car which is parked there.

And so on. This can go on endlessly, even if in real situations recursion will stop at a certain point, since the idea being expressed would get too confused. Recursion can also be applied to a noun and its adjectives:

Nice Alice.

And

Nice and cute Alice.

And again

Nice and cute Alice, sweet, gentle and considerate.

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Though, of course, recursion, like nasal consonants, is merely a very frequent (but not indispensible) property of natural languages, since Pirahã is known not to have recursive syntax. – jlawler Feb 23 at 5:15
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@jlawler: Pirahã is not known for its lack of recursion. This was claimed by Everett (2005), but it was seriously challenged by Pesetsky et al (2009). My point is that your comment seems to suggest that this is a fact, when this is actually a matter of debate in linguistics (although many linguists are fed up with it). – KleinePrins Feb 23 at 8:30
@KleinePrins It's not settled? I personally knew that Pirahã had been considered as a language that lacks it and so it challenged the theory that it's an inherent languages tract, an idea supported by Chomsky. – Alenanno♦ Feb 23 at 9:57
@Alenanno I don't know your source for personally knowing about Pirahã, but I think this is all part of a conceptual misunderstanding and an attack to a straw man. Chomsky didn't claim that all languages must share the same properties, but the critics seem to fail to acknowledge this. The idea is not that every language must be recursive, but that any human can learn a recursive grammar because recursion is a property of the Faculty of Language (not of languages). As for Pirahã, I think the reassessment by Pesetsky et al. is very convincing, but the data are still under consideration. – KleinePrins Feb 23 at 10:52
@KleinePrins There is no attack here, I was just reporting what I knew about it. But thanks for the info. :) – Alenanno♦ Feb 23 at 10:53
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