It's a trip usually done by newly married couples to celebrate their marriage. Now, the honeymoon is often celebrated in remote, exotic, warm, or other places that are considered special and romantic.
www.celebes.org The recent tendency among many couples is to combine marriage and honeymoon in an experience or replace one with another.
At first it did not have any reference to the one month period, but rather compared the love between the newly married people to the moon that was changing rapidly from full moon to fading away. Now it is usually a vacation that is passed together by newly married couples before they settle down at home.