The melody "Greensleeves" most likely originated as a dance tune during the 1500s.  It is among the most enduring of tunes and has appeared in many forms over the years.

The version you are listening to is taken from an 1855 English collection of music that had been popular in former days.  In its Christmas setting, a very early form of the text, as shown below, is found in a black-letter manuscript dating from the early 1600s, which notes that the accompanying music is "Greensleeves."

The Old Year Now Away Is Fled

The Old Year now away is fled,
The New Year it is entered:
Then let us now our sins down-tread,
And joyfully all appear!
Let's merry be this day,
And let us now both sport and play;
Hang grief, cast care away!
God send you a happy New Year!

The name-day now of Christ we keep,
Who for our sins did often weep;
His hands and feet were wounded deep,
And his blessed side with a spear;
His head they crown'd with thorn,
And at him they did laugh and scorn,
Who for our good was born:
God send us a happy New Year!

And now with New Year's gifts each friend
Unto each other they do send;
God grant we may our lives amend,
And that the truth may appear!
Now, like the snake your skin
Cast off, of evil thoughts and sin,
And so the year begin:
God send us a happy New Year!

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