English English (aka British English, King's/Queen's English) for most of its history did not employ widely standardised spelling. This changed in 1755 with the release of Samuel Johnson's Dictionary of the English Language, the first widely accepted standard dictioary.
During the American Revolution, British schoolbooks were obviously rare. A Connecticut teacher called Noah Webster (of Webster 1913 fame) decided to publish his own books. He published The American Spelling Book in 1788 (later to be called The Elementary Spelling Book) and over the following 40 years the book went through some 300 editions.
In 1828, he published the American Dictionary of the English Language, which is thought to have had a major influence on American spelling. Many characteristics of American English spelling were, however, already well established. "Center" and "theater", for example, were not new when Webster put them in his dictionary.
Webster believed in simplifying the spelling of words. He believed that children should learn to speak by pronouncing each syllable separately and clearly. Therefore silent letters such as the "u" in colour were discarded. Spellings such as "tough women" became "tuf wimmen". Webster recommended some pretty radical phonetic spelling, such as "soop", "fantom", "tuf", "hed", "medecin" and "tung". The dictionary sold very poorly, and Webster spent the rest of his life in poverty.
After Webster's death in 1843, Charles and George Merriam bought the rights to his dictionaries and published the first Merriam-Webster dictionary in 1847. It was this dictionary, not Webster's original, that became a runaway success all over America, and it left out many of Webster's "improved" spellings.
The main groups of words spelled differently on different sides of the Atlantic are those such as (american spellings) "theater", "civilize", "color" and "traveler". The -er and -ize spellings were fairly standard in the 1700s, but British writers (influenced by the French) were beginning to use -re and -ise endings. These were popularised by Johnson's 1755 dictionary, but they gained no foothold in the US. On the other hand, "colour" and "traveller" were the norm by 1700, but Webster rejected these in favour of "color" and "traveler", and Americans have followed his lead.
Probably the most famous difference in spelling comes from English chemist Humphry Davy's Alumin(i)um. He initially named it "aluminum", which is still used in America. The British, however, thought metals' names should end in -ium (i.e. magnesium, tecnetium), and changed the pronounciation as well.
Why the British havn't 'corrected' Platinum*, few can say.
*Or, indeed, Gold. Or Silver. Or Iron. Or Steel.
sloebertje says re American English: They probably didn't correct 'steel' because it's not an element :)
Well, yeah, that's true. I can't remember why I put that there, but I don't think I'd have dome it by accident.