What does Auld Lang Syne mean?

On New Year's Eve, the most common song for most English-speaking people to sing is "Auld Lang Syne." Isn't it funny how it's possible to sing and hear a song so many times and have no idea what it means? And wouldn't it be funny if it meant "Big Pink Elephants"?

A good sub-question is, what language is it?

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."