Your question refers to the song that Desdemona sings in Act IV scene 3 the night she is going to be murdered by her husband. What is important about this song is the way that Dedemona herself offers us the key to its symbolism. Note what she says about this song:
My mother had a maid called Barbary,
She was in love, and he she loved proved mad
And did forsake her. She had a song of "willow,"
An old thing 'twas, but it expressed her fortunes
And she died singing it.
What is important to realise is that this song possesses eerie echoes of her own situation, which is why it is so fitting to sing at this particular juncture. She too is in the position of worrying that the man she loves is losing is sanity and will leave her. In addition, let us remember that willows by water are traditionally a symbol of women who are left by their lovers. Consider how Ophelia in Hamlet drowns after Hamlet leaves her by falling out of a willow tree into water. The importance of the song therefore lies in the way that it allows Desdemona to express her own concerns and worries about her situation with Othello.