あけましておめでとうございます

• 謹きん賀が新しん年ねん

  明けましておめでとうございます is spoken (or casually written) year-round-appropriate greeting, while 謹賀新年 is a stiff, formal phrase used almost exclusively on New Year's cards.

• あけおめ

  明けましておめでとうございます is the full polite greeting, while あけおめ is its casual, shortened slang form used only with close friends.

• 明あけましておめでとうございます

  Standard kanji-kana mixed spelling; the most common written form.

• 明あけましておめでとう御ご座ざいます
  rarely-used kanji form

  Rare kanji form using 御座います for ございます; mostly seen in formal or traditional contexts.

• 明あけまして御お目め出で度とう御ご座ざいます
  rarely-used kanji form

  Very rare full-kanji form; archaic and decorative.

From 明ける (akeru, 'to dawn, to begin') in its continuative form 明けまして (akemashite) + おめでとう (omedetou, 'congratulations') + ございます (gozaimasu, polite copula). The phrase literally means 'congratulations on the beginning (of the new year)'.