New Years Greetings in Japanese

Since we are nearing the very end of the year, I thought it'd be very topical to give a detailed view over the various ways that you can in Japanese express felicitations for such occasion. And also, bask in the etymological joys of everything.

新年明けましておめでとうございま
Shinnen akemashite omedetou gozaimasu
This phrase literally breaks down as:
新年 (shinnen, "New Year")
明ける (akeru, "to open, to begin")
おめでたい (omedetai, "[a matter of] congratulations")
Here seen as おめでとう, an adverbial form typically seen in extremely polite circumstances, especially when paired with ございます (gozaimasu, "to be"), the honorific version of ある (aru ,"to be"), though to be more precise this actually reflects ござる (gozaru), which you may have heard/seen in samurai-related drama/comics - though it is originally the honorific form of aru.
Hence all in all, rather literally translated:
"Congratulations for the beginning of the New Year".
This greeting is only used after New Year has occurred, hence the "beginning" part. There are also informal varieties such as 明けましておめでとう (akemashiteomedetou) and the even shorter あけおめ (akeome), as well as merely:新年おめでとうございます (Shinnen omedetou gozaimasu).
Incidentally, 年明ける (toshi akeru) means "a (new) year dawns" is a related expression, using some of the same kanji.

謹賀新年
Kinga shinnen
Over in the more decorous side of things, this expression is only used in letters, and consists of 謹賀 (kinga, lit. "reverently (wishing) congratulations") - Used here as "expressing wishes of happiness", as well shinnen, which - as seen before - means "New Year."