This is one way of saying Happy New Year in Japanese. In English it is pronounced shin nen akemashite omedetou gozaimasu. Literally translated using Yahoo! Babel Fish, it reads "New Year Opening, you question with the [me], it is." Hmm, I don't think that helps much. Basically it means "Congratulations on the beginning of the new year."
This phrase, kotoshi mo yoroshiku onegai itashimasu, is very polite Japanese, which Babel Fish translates as "This year we ask may." Okaaaay. The concept of this sentence essentially asks the recipient of the New Year's card to continue being nice to the sender this year as in the past. We English speakers don't really have such a tidy phrase, but we should.