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Some languages have a "regulatory body" issuing recommendations and guidelines regarding the use of that language.

For example in the case of Spanish it's the Real Academia Española whose status is recognised in all Spanish-speaking countries. The Academy, among other things, publishes a dictionary ("DRAE"), in print and online, which is usually given a lot of prestige (but is not without controversies, of course).

Are there any such authorative—or at least influential—institution(s) or publication(s) for the English language?

Fortunately or unfortunately, no. There is no regulatory body like the Real Academia Española (or the Académie française, or the many others) deciding what is correct English; English evolves naturally with the changing usage of people. Right and wrong are decided based on describing and analyzing actual usage. (This — "descriptive linguistics" — is so canonical among English linguists that I've seen some of them occasionally find other languages' regulatory bodies an absurd idea. Anyway…)

Among dictionaries, the Oxford English dictionary and the Merriam-Webster dictionary are a couple of the "good" dictionaries (there are others), but note that these, too, have as their goal being reliable indicators of actual usage, and not regulation of, or authority over, language.

In the minds of most people, dictionaries and usage guides are a cipher to some presumed existing canonical, regulated definition of what is correct in the English language. Of course, no such canonical definition exists—grammaticality of English is governed only by the bulk of actual usage.

Most publishers of English dictionaries long ago abandoned any idea that they might set forth what is and is not correct in English—those few that actually did ever hold such a belief were few and far between. Modern English dictionaries, for the most part, are descriptive, although most do offer some degree of usage advice and notes. Merriam-Webster tend to be more descriptive than most, countenancing many usages criticized by others. The American Heritage Dictionary has its “Usage Panel” of experts on language and the usage notes in the dictionary cite percentages of the Usage Panel who approve or disapprove of questionable usages. The Oxford English Dictionary is widely revered as the canonical collection of English words, and it is certainly an amazing work of scholarly endeavor, most interesting in its coverage of historical English. But of course the OED holds no more official status than any other dictionary.

The most regulation we have today are style guides—such as the Chicago Manual of Style, the Associated Press Stylebook, and The MLA Style Manual. These are of course binding only on the writing governed by the producers of those style guides, but they are also used by many writers who are not required to follow them.

Then there are the professional peevologists, authors of such works as the Dictionary of Disagreeable English and The Big Book of Beastly Mispronunciations, who berate various usages and pronunciations they don’t like, citing whatever evidence supports their preferred usage or pronunciation, and ignoring the evidence that doesn’t.

Last, and certainly least, there is Strunk and White's The Elements of Style, perhaps the most overrated book on usage ever written, riddled with errors, hypocrisy, vacuous advice, and fatuous platitudes.

Summary

For what is or is not an English word:

Consider the current version of the North American Scrabble^® Players Association's Official Tournament and Club Word List and Long List as influential (and authoritative for Scrabble^® Tournament play) on the question of whether or not a word of 15 letters or less is an American English word. Outside North America, consider the World English-language Scrabble^® Players' Assocation's official list, called the Collins Scrabble Words.

For dictionaries considered authoritative:

Consider the dictionaries used to compile the Scrabble^® lists as very influential overall. See also this answer at US Equivalent to the Oxford English Dictionary which gives a different list.

For questions of Style and Usage:

Several organizations publish "style guides". For academic writing, the most influential are The MLA Handbook for academic writing at the high-school and undergraduate level, The MLA Style Manual for more advanced scholarly writing, The AP Stylebook for news and journalism, and The Chicago Manual of Style for general purposes. Also, The American Heritage Dictionary is notable for its use of a "Usage Panel" of 200 "prominent users of the language" which provides some of the best guidance available for what is considered best usage of words and word forms among controversial alternatives.

While there is no offical regulatory body for the English language, one influential regulatory body for American English is the North American Scrabble^® Players Association. They provide a list of words acceptable for use in Scrabble^® play, and thus judge whether or not a word is a word.

To be fair, they acknowledge that their list is not a complete list of English words, excluding words no longer in use among other things. Also, because of the nature of the Scrabble^® game, they do not address words longer than 15 letters. Still, this is, at least in my mind, the most authoritative list of American English words there is. Which is not to say that OED is wrong if they include a word not on that list, but rather that if a word is not on that list, I would not consider it a current American English word.

A few things to note about the Scrabble^® lists:

• 
  The Official Scrabble Players' Dictionary (OSPD) is meant for use in children's tournaments, and therefore excludes offensive words such as "asshole". It is not a complete list even by Scrabble^® standards. As of this writing, the current version is OSPD4.


• 
  The Official Tournament and Club Word List (OWL) only includes 2-9 letter words. As of this writing, the current version is OWL2, with the next version expected in 2014. It is also focused on usage in the United States.


• 
  The Long List is a supplement to OWL2. It contains 10-15 letter words.


• 
  The Collins Scrabble Words list (CSW) is a superset of OWL2 that adds words (and spellings) used outside of the US. The current version is CSW12

At the moment, you can access electronic versions of these lists via the free study program Zyzzyva. The lists from Zyzzyva include cursory definitions of one sense of the word. The definitions are provided only to satisfy basic curiosity about the word, not to be in any way comprehensive.

On the subject of authorities, it is worth noting what authorities were used to compile the Scrabble^® word lists. OWL2 was compiled with reference to:

• American Heritage College Dictionary (4th edition),


• Merriam-Webster’s Collegiate Dictionary (11th edition, 2003 printing)


• Random House Webster’s College Dictionary (2nd revised and updated edition, 2000)


• Webster's New World College Dictionary (4th edition)

The Long List is published by Merriam-Webster, so it is perhaps not as surprising that it was based on Merriam-Webster’s Collegiate Dictionary, 10th edition

The CSW adds words from Collins and Chambers dictionaries to the current version of the OWL.

Google, Apple, Microsoft, and Amazon Kindle all license OxfordDictionaries.com, AKA Oxford Living Dictionaries, previously known as Oxford Dictionaries Online (ODO), for their definitions. Specifically, the Oxford Dictionary of English (ODE) and the slight variant, the New Oxford American Dictionary (NOAD). I have another question clearing up confusion about all the various Oxford dictionaries. The OED, of course, stands apart from all of them, despite sharing a publisher (Oxford University Press).

I believe these companies have lent Oxford Dictionaries (ODE/NOAD) a massive air of authority, both in terms of their endorsement and just by the sheer ubiquity that these definitions now have.

(I'm actually pretty upset about this because I've gradually become convinced it's an inferior dictionary with a lot of missing and incorrect (in the descriptivist sense!) definitions.)