The rose bush symbolizes a variety of people, reading enotes style section attached below will give you a great variety of those.
For a most basic symbol, it represents Hester. The rose is a beautiful flower that grows and withers. It has great fragrance and these like positive qualities can be found in Hester with her great skill and humble personality. But a rose will stand up for itself as Hester eventually asserts herself in the novel to Chillingworth and the magistrates. Hester's life is both beauty and pain, like a rose and its thorns. The red color of a rose has positive and negative connotations as does Hester's love. First, her love was with a forbidden man, thus the negative. Next, her love was a pure and true love, unlike the love of her real marriage. Many more parallels can be drawn.
Hope this helps you.