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.
The most important symbol in the novel Harry Potter and the Sorcerer's Stone is the scar on Harry's forehead. The scar represents Harry's past, his present, and his future.
The scar is representative of Harry's link to Lord Voldemort. Voldemort is the one who gave Harry the scar as a young child when he murdered Harry's parents. The scar forces people to become instantly drawn to Harry based upon the prophecy of his rise.
Another symbol in the novel is the names associated with each of the different houses at Hogwarts. Perhaps the most telling house is that of Slytherin. The image of the house is that of the snake. Historically, as far back to Adam and Eve, the image of the snake represents evil.
One last symbol is the game of Quidditch. The game not only brings pride to each house, it represents the importance of values.
one symbol is the broom... J.K. Rowling could have choose any other object or even an living creature and made the book much more interesting but he chose not too... so there is a reason behind this which is personal to him