This is traditionally translated as “Happy New Year”, but I like the translation Steven L. Renshaw provides on his “New Year in Japan” page:
Japanese express wishes for the New Year by saying “Akemashite Omedetou Gozaimasu!” (pronounced ah-keh-mah-shteh oh-meh-deh-toe go-zah-ee-mahss). Only one Kanji (Chinese character) is found in this phrase (within the first word). This Kanji is a combination of the characters for sun and moon, and among other ancient meanings, it has to do with the sun and the moon getting together and becoming “bright”. It entails “changing” and “opening”… “dawning”…
In ancient lore (under the lunar calendar), the New Year was seen in relation to change in both the sun and moon as well as the symbolism of their luminance. The meaning(s) of the phrase “Akemashite Omedetou Gozaimasu” may be somewhat complicated, but (roughly translated) may include the following: “The year is changing… darkness gives way to light… new life begins… Congratulations!”
Interestingly, though he doesn’t go into this, all of the “changing” part of the meaning is embodied in “akemashite”; “omedetou gozaimasu” is “congratulations”, which is also used in “tanjoubi omedetou gozaimasu” (Happy Birthday) and other congratulatory phrases. So, that’s a lot of meaning in one small word, most of it brought through the kanji 明.
(I found this New Year’s explanation via Sid’s New Year’s entry.)