Somber and sombre are different spellings of the same word, meaning (1) dark and gloomy, or (2) melancholy. Somber is preferred in American English, while sombre is preferred in all the other main varieties of English.
Sombre is the original, taking the spelling of the French word from with the English word derives. The more American spelling arose gradually through the late 19th and early 20th centuries until becoming the dominant form in American writing around 1940. This brought somber in line with the numerous other words whose original -re endings had become -er much earlier in American English. Centre, for instance, gave way to center in the U.S. by around 1910.
Outside the U.S.
Joy was met with sombre remembrance as 42 fresh faces entered the Toronto Police Service Thursday. [Toronto Sun]
Derek Cianfrance’s film is a sombre, painful portrait of a toxic marriage. [The Guardian]
Jazzing up subject titles may help revive interest in Indonesian studies, an otherwise sombre paper on the decline of the discipline reports. [The Australian]
Those events complete a somber week at the White House. [Washington Post]
Forgive the somber tones this week, but February is always a tough month for me. [Wilkes Barre Times-Leader]
Everything he touches comes out sounding somber and a bit grandiose. [San Francisco Chronicle]