shedding some light on the myth
So often -- particularly post-movies, but it did occur before they were released -- in the Tolkien fandom, Arwen is referred to as an 'elven princess'. Now, while she is most certainly elven, the truth of the matter is that she is not actually a princess!
The mistake is forgivable, however. We are introduced to Arwen as the daughter of an elven lord, the closest thing to royalty that the elves have left in Middle-earth. Her father was the herald to the last elven king east of the Sundering Seas, Ereinion Gil-galad. Elrond also was something of a scion to Gil-galad, in the fact that Vilya, elven ring of Air, was left in his keeping by his lord. We also learn that she is the granddaughter of Galadriel and Celeborn of Lothlórien, the Lord and Lady of the Golden Wood. It's quite an easy conclusion to jump to. But though Arwen is descended from royalty (and from holy beings!) she is not a princess herself. Never in his writings does Professor Tolkien give Arwen the title of 'princess' -- before becoming Queen of Gondor, she is referred to as Lady Arwen, the Lady of Rivendell, and so on.
It has been argued that being the granddaughter of Galadriel and Celeborn makes Arwen a princess of Lothlórien. This is not so, as Galadriel and Celeborn are not the King and Queen of Lórien, but the Lord and Lady. They never took up the position of royalty after King Amroth forsook the Golden Wood to take his beloved Nimrodel to the Blessed Lands.
Going back through her ancestry, we can trace the royalty in her blood. Arwen is directly descended from Queen Melian and King Thingol of Doriath (whose daughter was the famous Lúthien Tinúviel), also from King Turgon of Gondolin, the Hidden Kingdom. In fact, Arwen's royal blood goes directly back to one of the very first elves to set foot in the blessed lands of Valinor -- Finwë, High King of the Noldor. While this is an impressive bloodline to boast, the kingdoms of her royal forebears had long since collapsed by the time the events in The Lord of the Rings took place, and you can't be a princess of something that no longer exists!
While Arwen is not a princess, she is still highly revered and loved by her people as the last glittering example of elvenkind in a fading age.