日曜日(nichi-youbi) literally translates as sun-day
月曜日(getsu-youbi) is moon-day, or Monday
(the bi bit is the part used for sun, one sun being a day, naturally)
I couldn't understand how this might have occurred unless by Japan adopting new names for the days of the week, and presumably a western style seven day week along with all the other things it took up during Meiji.
Well, as it happens the story is more interesting. The Japanese calendar is, surprise surprise, based on the Chinese calendar. The Chinese calendar is itself the product of a Babylonian influence on pretty much the whole world. The Babylonian seven day week, probably born from the 28 day lunar cycle and the need by early agrarian societies to meet to trade, was named after the seven bodies in the sky that were visible, ie the sun, moon and five planets.
In English this got corrupted as certain days of the week were re-associated with Norse gods, for example Thursday = Thor's Day, Friday = Freya's Day; but the sun and moon remained.
In Japanese, from Chinese, the relationships for the rest of the week are:
火曜日 (ka-youbi) or fire-day = Tuesday
水曜日 (sui) or water-day = Wednesday
木曜日 (moku) or wood-day = Thursday
金曜日 (kin) or metal/gold-day = Friday
土曜日 (do) or earth-day = Saturday
The observant might spot that these relationships are like the elements, and that's about it; in Chinese mythology the planets are associated with elements - for example the planet Saturn is associated with the element earth. Which is, lo and behold, found to be doyoubi, or Saturn's Day = Saturday.
On a related note, in written form the days in Japanese usually omit the you bit of the days of the week, so 土日、日日 etc would be common. This 曜(you) bit is a little odd as while in Japanese its use is limited to the meaning discussed here, in Chinese it means 'pretty bird' or something along those lines.
NOTE: this is predominantly pieced together from Wikipedia (here and here), which as we all know is an entirely unreliable source, but as the facts seem to hold together from various entries, and bare out a logical consideration of the facts, I'm content that in this instance it seems reasonable.