The story behind the Ganesha head is that Goddess Parvati created Ganesha without the knowledge of her consort, i.e., Lord Shiva. When Lord Shiva comes to meet Goddess Parvati, Ganesha refuses to allow him to do so. Shiva beheads Ganesha without knowing that he is his son. When Goddess Parvati retaliates, then Lord Shiva replaces the head of the Ganesha with that of a young elephant.
So, what does the elephant head of Ganesha signifies? If an obstacle in the path of an idea needs to be removed then the same must be done at the stage of daydreaming and thus, Lord Shiva decides to go with the elephant head. It signifies that the obstacle created by the idea must not get converted to action without giving due consideration to the process.