In the eleventh century, William the Conquerer from France defeated the English King Harold, and brought the French language to England. English went into a bit of a decline because all official government writing was in French (so between that and church-related stuff in Latin, very few people wrote in English). and since French was the hot new thing in town, people began adopting many French words into English (most of which relate to government or conquering, such as "soldier," "traitor," and "governor"). So, English (by this point it was Middle English) was still the vernacular, while French was the language of high society. but still, neither one had standardized spelling, and you could spell stuff however you wanted as long as your intention got across.
In the fifteenth century, Geoffrey Chaucer made English popular again with The Canterbury Tales, and the language had a huge resurgence after the printing press was invented in the sixteenth century, since it was suddenly so cheap and easy to print whatever you wanted. However, printing presses typically came with just Latin characters, so all the extra runic characters in English had to be discarded (they had already begun to fall out of favor, but this was definitely the end for thorn, edh, yogh, and wynn; ash and oethel were dropped from the language later on, since they could still be typeset using the Roman alphabet). There needed to be a way to represent these letters that couldn't be typeset, so people improvised: thorn and edh both became TH, wynn became W, and yogh was in an unpleasant position because there isn't a good way to substitute letters for it. It's now variously written as a G, Y, GH, or W (Chaucer rhymes "plow" and "enough" because they both ended with a yogh back then; similarly, Pittsburgh, Salisbury, and St. Petersburg all end in the same root, along with the burroughs of New York City). This was the beginning of Modern English. Still, spelling was somewhat fluid, and words were often spelled different ways within the same book because the typesetter didn't remember how it had been spelled 50 pages back, and just wanted to print the book as quickly as possible. However, spelling had begun to crystallize because you could compare spellings from different printers and pick the one you liked best and use it in the future.
Even though spellings were starting to standardize, they had changed a lot since Olde English (heck, even the E in Olde had been added; it wouldn't have been spelled that way by King Alfred). In an effort to make things seem more Latin-like, people started adding in silent B's to "debt" and "lamb." "Shew" swapped out its E for an O. "Wyf" traded the Y for an I and added a silent E to the end to signify that the I should use the long pronunciation. As the British empire spread, new words were brought into English from other languages, including "jodhpur", "chipmunk," and "algorithm." Modern English is roughly 90% borrowed words, and only 10% derived from the original Germanic roots.
People then started wondering whether we should standardize spelling, since it was no longer so rare to see a book that had been written hundreds of miles away (and who knew what sort of pseudo-Latin humbug had been inserted into those crazy words?). By the eighteenth century, people had begun making dictionaries, though they were often very short collections of unusual words to learn about, rather than something to show elementary schoolers. Samuel "Dr." Johnson (I have no idea if he was a doctor or had any degree; people called him Dr. Johnson, as a sort of nickname) finally published the first comprehensive-ish dictionary of English in 1755. It was this dictionary that forever changed spelling from a descriptivist (spell the way that seems most useful) to a prescriptivist (spell the way it's written in the dictionary) sort of practice.
Once the dictionary took root, it became very hard to create an alternative spelling. A few people managed to do it, though. Chief among them was Noah Webster, who decided that even though America had won its independence from England, it was still too attached to English culture. He therefore made his own dictionary, in which any spelling that could plausibly be changed was modified as a way to break further ties to the language's British heritage. The U was dropped from "color" and "honor," the R and E were switched in "theater," "grey" changed its vowel, "critick" dropped its K, "apologize" got a Z instead of its previous S, etc. However, there were still an annoyingly large number of historic spellings and unexpected silent letters.
Throughout the past few centuries, other people have wished to further revise spelling in order to make it easier. and these aren't just wackos and kooks; I'm talking about Benjamin Franklin, Melville Dewey (creator of the Dewey Decimal System; he changed his name to Melvil Dui until the outlandish spelling started holding his career back), Andrew Carnegie, Mark Twain, Theodore Roosevelt, Ronald Reagan, the head of the Chicago Tribune (back in the '40s; they caved in to popular spelling a few decades later) and various presidents of various colleges and universities like Cornell and Harvard.
Unfortunately, it's no longer feasible to revise English orthography. If you want to make it phonetic, you'll have to endorse one accent over all the others (for instance, should "schedule" be "skedjool" or "shedjul," as they're both commonly used pronunciations in different parts of the world). The problem here lies in authority: who am I to decree that the way everyone has been spelling for the past 200 years is wrong? Then again, going the anti-authority route is similarly unlikely: if I start yewsing mai own speling, peepil wil think I'm dooing it rong and trai too korekt me. It's also partly a class thing: if you spell correctly, it looks like you are educated, and this has been used in the past to deny opportunities to immigrants and people who can't afford a good education. Also note how LOLcats captions are often perceived as depicting uneducated cats (I'm pretty sure there have been LOLcat captions where the cat says something like "I disagree with the general premise that my felinity precludes correct orthography,"
and yet, orthography reform is happening naturally right this very moment. Tho its often relegated to teh intarwebs and TXT messages, enuf people R using new spellings that their becoming part of the vernacular despite there so-called "incorrectness." We have similar spelling modification in the names of businesses like Toys 'R' Us, Qwest, and Google (should be googol). and this goes right back to the original philosophy in Old English: it doesn't matter how you spell as long as other people can understand what you're trying to say.