The spirit Ariel is the cause of any unusual music in the play, for example in Act 3/ Scene 2 when Ariel plays his flute as Caliban convinces Stephano and Trinculo to assassinate Prospero and take over the island. I don't think a particular song relates to anything much but the music that Ariel plays throughout in general could be related to certain themes.
Music relates to themes of reality - what we percieve to be real. Ariel plays it when he is invisible and so the other characters do not know where it is coming from or that he even exists. This information is only privy to Prospero and us, the audience. Therefore everyone but Prospero has a warped idea of what the reality of the music is... if that makes sense? Ariel is also very much related to themes of human perceptions of reality as he is ultimately the catalyst (via Prospero) of all supernatural occurences in the play, e.g. music when Sebastion and Antonio plot Alonso's murder, the banquet, clothing, the harpies and remains unbeknowst to all the other characters.
If you use a post-colonial reading of the play you could also use the music in Act 3/Scene 2 to describe Caliban's character. From a post-colonial perspective he can be seen as an oppressed character possibly symbolic of other oppressed people such as the black native Americans who, although indigeonous to the land were made slaves. Shakespeare mostly portrays Caliban as a hateful savage whose "nature, nurture could never stick" but occasionally the audience is allowed to glimpse his more sensitive side. This occurs as Ariel plays the music when he is with Stephano and Trinculo. It is interesting because although the other two are deemed 'human' it is Caliban who appreciates the music more and describes its beauty. Music is a form of art and because Caliban can understand it we are made to question if he really is just a "monster" or something more. This is what helps make him such a complex character on closer analysis.
A good website is: http://www.sparknotes.com/shakespeare/tempest/
Well, at certian parts of the play, Ariel uses magic to lure some of the castaways on the island. In Act I, Scene II, Ariel sings the "Come unto these yellow sands and then take hands" song to Ferdinand. And then several lines later Ariel sings the "Full fathom five they father lies..." song to warn Ferdinand of where his father is (Ariel says he is at the bottom of the ocean, but we know that he is actually safe and on the island). Then later in Act II, Scene I, Ariel plays a song on a pipe to put the court to sleep, but then sings a deifferent one to wake them. Later in Act 3, Scene 3, the nymphs enter to music with a banquet to tempt the court (Alonso and company) into eating posionus food, but then Ariel enters dressed as a harpy and scare them. In Act IV, Scene I the godesses (Ceres, Juno, and Iris) sing the "Honor, Riches, Marrige, Blessing..." song to ferdinand and Miranda to celebrate their love. And lastly, in Act V, Scene I, Ariel sings the "Where the bee sucks..." song to Prospero as she is putting on her garmets simply to add music and make Prospero's life more enjoyable, as she does in every scene, because she is such a loyal servant (Prospero sets her free in the end of Act V).