Beren and Luthien were figures in Tolkien's Silmarillion, and fell in love during their fight against the dark lord Morgoth. However, Luthien, who was "the most beautiful of all the Children of Ilúvatar", was an elf, the daughter of the King and Queen of Doriath. Luthien's father did not wish for them to marry, and set the condition of marriage to be the returning of a Silmaril from Morgoth's crown. In the act of completing it, Beren was killed, and Luthien lay down and died of grief. Eru himself offered two choices:
she could either dwell in Valimar with the Valar in bliss forever as reward for all that she had accomplished, or she could be restored to life again with Beren, on the condition that they would both be mortal and die the death of Men. For her love of Beren, Luthien chose the latter.
Because of this, their children would become half-elven, being able to choose either a human or elvish fate. As fabikw notes in his answer, Elrond, who was one of these descendants, chose to become an elf, was Arwen's father, allowing her to make a choice herself. Aragorn was a descendant of Beren, but from a side that chose the human fate. Thus, when they met millenia later, they were a picture of Beren and Luthien. Arwen ultimately made the same choice Luthien did, and became mortal and eventually died (possibly of a broken heart).
So yes, this song has everything to do with Arwen, as it is both a history of her ancestors, as well as a foreshadowing of her life. Arwen was also noted to be very beautiful as her ancestor Luthien was. As an aside, see here for a brief explanation of the significance of Beren and Luthien to Tolkien.
additional note: Aragorn wore the Ring of Barahir as a token of his lineage. Barahir was Beren's father