One is the correct spelling in the UK - coloured and the other is the correct spelling in the USA -colored. So they are each correct however if you used the spelling appropriate in each country in the opposite to which it was normally used it would not be correct. In UK colored would not be correct and in USA coloured would not be correct. It is just a peculiarity between them.

Colored (without a u) is the US spelling, and coloured is the Commonwealth (UK and related countries) spelling. The important thing is to be consistent. You can write "my favorite color" or "my favourite colour", but "my favorite colour" and "my favourite color" are simply messy.

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BOTH are correct. If British, Irish, Australian, New Zealand or South African people see 'colored', they don't think 'That's wrong'. They think, 'Oh, that must be an American text' and then just read on quite happily. Likewise if Americans read 'coloured', they don't consider this incorrect, just simply British, and then they'll ignore it. The Canadians are nice laid-back folk and they're happy with any spelling convention. Editors and spellcheckers (both automatic and human) might amend spellings to fit local conventions, but neither form is ever considered 'wrong'. I'm an English language examiner for a major exam board, and take it from me: both are accepted as equally correct. As Peachey recommends, you need to be consistent - it doesn't look good if you swap between spellings within once text - but otherwise you needn't worry about which one to use.