What does John 2:1 mean?

"The third day" probably means the third day after Jesus recruited Philip and Nathanael (John 1:43–51). It probably doesn't mean Tuesday; in the Jewish tradition, virgin brides married on Wednesdays. Widows married on Thursdays. In Jesus' era, weddings were not single-day events. The realities of travel in that era, for messengers and guests, made it difficult to predict exactly when a person would arrive. A typical celebration could last a week.

We don't know specifically where Cana was, but it was the hometown of Nathanael (John 21:2). Nathanael might have been the original invitee, for that reason. It's possible that Jesus' mother, Mary, was helping to serve guests at the wedding, explaining her involvement in the story (John 2:3). The wording between this and the following verse (John 2:2) suggests Mary was already there and Jesus and His disciples came later.

Weddings are a common source of symbolism in Scripture; God is often compared to a groom, and His people compared to a bride.