From Tolkienonline, dated March 27:
Here's a rundown of the new scenes you can look for:
*A new addition to the opening sequence in which Bilbo provides background on Hobbits and their history in voice-over as he writes his memoirs.
*A new introduction to Samwise Gamgee, seen in his capacity as a gardener.
*A scene taking place at the Green Dragon Inn, which introduces us to the camaraderie of the Hobbits (we see them singing together) and sets up the geopolitics of the story.
*The Hobbits witnessing the departure of the Elves from Middle Earth on the way to Bree.
*Aragorn watching over the sleeping Hobbits, singing the ballad of Beren and Luthien to himself in the night.
*Aragorn at his mother's grave, in which we learn that he was raised by Elves and that Sauron has long hunted him.
*Two new moments during the departure from Rivendale, one in which we see Arwen's emotional reaction to Aragorn's leaving, and another in which Elrond sees the Fellowship off.
*A scene with the Fellowship in the mines of Moria, in which we learn how the Dwarves themselves unleashed the fire-demon that eventually destroyed them.
*A scene at Lothlorien, where Galadriel bestows upon each of the Fellowship a gift which will play an important role later in the Trilogy.
*And finally, more footage of the battle at Amon Hen.