Favor and favour are different spellings of the same word. Favor is the preferred spelling in American English. Favour is preferred in all other main varieties of English. These preferences extend to most derivatives, including favored/favoured, favoring/favouring, favorite/favourite, and favorable/favourable.
The American adoption of favor was part of a broader early-19th-century effort to create a distinctly American English with spellings considered tidier or more logical. Favour was just one of many similar words that lost the u during this time. Others include colour (now color in the U.S.), labour (now labor), and rumour (now rumor). There are many others. The ngram below, which graphs the occurrence of favor and favour in a large number of American texts published between 1800 and 2000, shows that the shorter spelling gained ascendance in the U.S. in the 1840s:
Favor is now the more American spelling, but what is often forgotten in discussion of these things is that favor is not an American creation. The word’s spelling was in flux through much of its history and wasn’t permanently settled even in British English until the 18th century. The Oxford English Dictionary cites instances of favor from as long ago as the 14th century, and it appears along with other early variants such as fayuer and fauour. Americans didn’t invent what is now their spelling; they just chose an old form that was different from the one then preferred by Britons.
Outside the U.S.
Punters have written off Julia Gillard in favour of former union boss Bill Shorten and backed the coalition to win the next federal election. [The Age (Australia)]
And that means Ferguson will urge Birmingham City to do him a favour against Arsenal. [Guardian (U.K.)]
The Supreme Court evidently favours a co-operative approach. [Globe and Mail (Canada)]
The opposition Congress twice staged a walkout from the Chhattisgarh Assembly today over the alleged favour shown by the state government. [MSN India (dead link)]
The intent of a system that favors the youngest and the healthiest is to ensure the best and longest use of donated kidneys. [New York Times]
71% oppose increasing sales, income or other taxes while 27% are in favor that approach. [USA Today]
Torres knew Chang was associated with the Giants but did not know he was an owner and asked for a favor. [San Francisco Chronicle]