No single authority decides what is right or wrong in English grammar. English has no official language academy: its rules are conventions that dictionaries, style guides, publishers, and teachers record, and in the end, usage decides.
Every writer asks this question once in a while. Whenever people discuss style and grammar, someone is sure to ask the decisive question: "On what authority do you say that this is an error?"
Leaving aside what we would all call obvious errors, this is an important question. There are conventions we all learn at school and connect with correct written English.
Everyone recognizes a typo, a misspelling, or a serious grammar mistake. But beyond the obvious, it gets tricky, because there is no higher authority that defines right and wrong once and for all.
No one rules English
The world is full of defined rules, from standard paper sizes to the quality classes of bananas for export. Grammar and spelling rules, by comparison, are islands of convention in a sea of rules and exceptions to those rules.
There is no official body for English, the way the W3C sets the standards of the web. Yet many spell checkers and grammar checkers apply rules, and so in a way they also decide what proper English is.
On top of that, English is spoken and written differently in different places. British, American, Canadian, Australian, and the English of African and Asian countries differ in spelling and usage. (Here is how to keep British and American spelling consistent.)
So who rules? Does Britain have a historical right to set the grammar rules? Does the United States have the political weight to impose its own? Not really.
However, some institutions have enough authority and influence to establish a set of rules that most people accept as "right". These rules are the basis for spell checkers, editors and proofreaders, teachers, and universities.
Some of them carry more prestige and therefore more weight, such as the big dictionaries (the Oxford English Dictionary, Merriam-Webster) and the style guides of publishers, universities, and news agencies, like The Chicago Manual of Style or the AP Stylebook.
How the rules came about
As in many European languages, the rules of English grew with its history, and mostly along with the development of print.
English grammar and spelling rules came out of necessity. There were many regional dialects, and people from one region often could not understand people from another. When printed books arrived, communication needed some standardization.
The first printed grammar books of English date from the late 16th century, the time of the Renaissance. Other European countries saw the same trend.
As always, those who had the power made the rules. What we call standard British English today grew out of the English spoken in London and the East Midlands. The prestige of Oxford and Cambridge as centers of learning helped make it the standard written English.
If history had gone differently, standard English might be based on a northern or Scottish variety and be taught and spoken quite differently today. But unlike French or Spanish, which have the Académie française and the Real Academia Española, English never got an official language academy. As a result, English grammar is a set of rules that many people have agreed on, with no institution in charge.
Usage decides
This leads to a pragmatic and flexible use of grammar. English is very open to new words and to elements of spoken language finding their way into accepted usage. Some rules that people still defend were never real rules at all; see 2 common grammar errors that might not be errors.
That is a good thing. Language is a living thing, shaped by everyone who speaks and writes it, and it should stay that way.
In the end, proper grammar is decided by usage within a community. You can see this in the language settings of any spell checker, which offer separate versions of English for different regions. The Spell Checker, for example, lets you choose American, British, Australian, Canadian, New Zealand, or South African English.