American English has both spelling and grammatical differences from British English, some of which were made as part of an attempt to rationalize the English spelling used by British English at the time. Unlike many 20th century language reforms (e.g., Turkey's alphabet shift, Norway's spelling reform) the American spelling changes were not driven by government, but by textbook writers and dictionary makers.
The first American dictionary was written by Noah Webster. At the time America was a relatively new country and Webster's particular contribution was to show that the region spoke a different dialect from Britain, and so wrote a dictionary with many spellings differing from the standard. Many of these changes were initiated unilaterally by Webster.
Webster also argued for many "simplifications" to the idiomatic spelling of the period. Somewhat ironically many, although not all, of his simplifications fell into common usage alongside the original versions, resulting in a situation even more confused than before.
Many words are shortened and differ from other versions of English. Words such as center are used instead of centre in other versions of English. And there are many, many other variations.
American English has further changed due to the influx of non-English speakers whose words sometimes enter American vernacular. Many words have entered American English from Spanish, etc.
Examples of common American English borrowed words, probably not common in British English (except for kosher, patio, pronto, incommunicado, etc.).
From Yiddish
From Native American languages
From Spanish
From Dutch
From African languages
Origin Unknown
For detailed (and now mostly correct) differences in British English and American English see American and British English Differences.