The Dursleys, Harry's Muggle guardians as the novel opens, symbolize narrow-mindedness, conventionality and lack of imagination. They fear what they know of Harry, and do their best to keep him down and oppressed. They are incapable of his vision or of entering the magical wizarding world. They are the forces in a society that would, out of fear, destroy its creative qualities and potential for change.
The magical world that Harry joins, on the other hand, symbolizes the numinous and the imaginative, a universe of possibility open to those with eyes to see it, lurking in the interstices of the known world. Platform 9 and 3/4, for example, is hiding in plain sight, as is Diagon Alley, but only those with the proper vision can perceive these places. The ordinary Muggle passes blindly by a sparkling other world, not conceiving it's existence. Harry's wizarding world could be likened to "the more things in heaven and earth" that Hamlet says are missing from Horatio's materialist philosophy. It reminds us to be aware of the beauty and possibility all around us that we might otherwise miss.
On a more minute level, even the materials used to make the wands are symbolic. For example, Harry's wand contains a phoenix feather, a symbol of rebirth. This symbolizes that Harry, the despised child in the Muggle world, has been "reborn" to a heroic role in the alternate, magical world he enters.