Have you ever wondered what they are singing and what it's about when we sing "Auld Lang Syne"? I found a clever article about this. I had to laugh because I never sing the song correctly. Here's the article and the wikipedia page about the song.
It turns out that "Auld Lang Syne" is an extremely old Scottish song that was first written down in the 1700s. Robert Burns is the person whose transcription got the most attention, so the song is associated with him.
According to this page, a good translation of the words "auld lang syne" is "times gone by." So (incorporating a couple of other translations) when we sing this song, we are saying, "We'll drink a cup of kindness yet for times gone by."
Here's a whole list of New Year's songs from Beliefnet...who knew?