For the noun referring to an abnormal tissue growth, tumor is the preferred spelling in American English. Tumour is the standard spelling outside North America. Canadians likewise prefer tumour by a large margin, though tumor does appear in Canadian publications with relative frequency.
Tumour is one of numerous -our-ending words that lost the u in American English at some point during the 19th century. In this case, the u was dropped for good by around 1880, though tumour has never fully disappeared from U.S. writing, and in fact a few medical publishers still prefer it.