Critical Period Hypothesis
The critical period is defined as being the “biologically determined period of life when language can be acquired more easily and beyond which time language is increasingly difficult to acquire.”
The critical period hypothesis is a scientific argument based on biological facts on how the human brain develops and changes over time. Before puberty, the brain has high plasticity—not in the sense that you could mold McDonald's toys out of it, but in that its development is very flexible, with an extraordinary ability to absorb new information and concepts.
After puberty, the hypothesis goes, the brain begins to undergo a process called lateralization, in which certain functions, like language, are assigned to either the left or right hemisphere of the brain, which would make further language acquisition difficult. A good deal of this is theorized to be because people will actually begin to overanalyze any attempts to learn a language, as opposed to learning in an organic and open manner.
As with all things scientific in nature, this hypothesis is up for debate. Does it hold while under scientific scrutiny?
Well, it's not a simple answer, as with most things. There are two main approaches to CPH, an extreme view and a softer view. The extreme view, that after puberty true language acquisition is impossible, has largely been negated. However, a softer view is that language acquisition is merely more difficult and not necessarily impossible. One aspect where CPH has held particularly strong is with regard to phonology in language acquisition–basically, the ability to develop a native accent.
There is a lot of interesting research going on right now with regards to CPH. For instance, check out this recent study by Lauren Cox of the ABC Medical News Unit in which babies are found to cry in their parent's accents by their second day of life. The validity and strength of CPH has been a longstanding debate for decades among psycholinguists and appears set to continue for a good deal more.
Even if some tenets of the critical period hypothesis are true, there is a huge amount of individual variation when it comes to language acquisition. Some people have a natural talent and can pick up a half dozen languages with ease throughout their entire life; others have a rather lot of difficulty even with their first. While statistics and science may serve to illustrate general trends, they need to be remembered as such: general trends. People learn different things in different ways with differing levels of difficulty: it's a fact of human nature, so it's difficult to generalize in something as complex as language acquisition.
Not Quite Fluent
An important consideration of CPH: It applies most critically to accent and less to vocabulary and grammar. So, while you may not achieve perfect fluency, enough hard work can and will get you somewhere in a language, even if it's just limited to navigating a grocery store in a foreign country or ordering your coffee in another language or reading the newspaper. Most people who learn a language in later life, even if they have picture-perfect grammar and use of idiom, will retain some foreign accent despite all their best efforts. This doesn't necessarily mean that learning a language later in life is worthless, just that you aren't going to sound strictly native.
How’s That Sound?
Yes, it's true, it's easier to learn a language when you're a young prepubescent child, and this has a fairly strong scientific (and anecdotal) basis. However, just because you may be well past puberty does not mean that you should refrain from learning a language. It can still be an incredibly worthwhile activity, whether you're doing it for travel, for business, or for the sheer pleasure of becoming intimately acquainted with another culture.
For more information, I highly recommend the Wikipedia article on the critical period hypothesis. It has extensive citations to the scientific literature, those supporting, negating or otherwise complicating the question of second language acquisition, as well as a more in-depth overview of the subject.
For an interesting read on CPH, check out the book Second Language Acquisition and the Critical Period Hypothesis.