From the Tolken's LotR series it is mentioned that the Sword of Elendil was forged anew before Frodo departed from Elrond's house, but if my memory serves me correctly, in the Peter Jackson's rendition of the LotR movie, it was reforged at the end of The Two Towers, after Saruman was destroyed.
My question is, am I right with regards to Peter Jackson's movie? When in the movie was the Sword of Elendil reforged?
You are correct on both counts; in the movie Arwen requests that her father Elrond reforge the sword and he brings it to Aragorn at the Rohirrim muster at Dunharrow just before he sets off to the Land of the Dead.
In the book it is forged before they depart from Rivendell:
The Sword of Elendil was forged anew by Elvish smiths, and on its blade was traced a device of seven stars set between the crescent Moon and the rayed Sun, and about them was written many runes; for Aragorn son of Arathorn was going to war upon the marches of Mordor. Very bright was that sword when it was made whole again; the light of the sun shone redly in it, and the light of the moon shone cold, and its edge was hard and keen. And Aragorn gave it a new name and called it Andúril, Flame of the West.
As covered in the accepted answer the shards of Narsil were reforged into Anduril while the original fellowship was still being formed in Rivendell.
Another departure between the book and the film is that, in the book Aragorn carried, at least the hilt of Narsil with him up to that point.
From 'Strider - FOTR' pp168 (2001 edition) : the chapter where the hobbits first meet Aragorn at the Prancing Pony in Bree.
'He stood up, and seemed suddenly to grow taller. I his eyes gleamed a light, keen and commanding. throwing back his cloak, he laid his hand on the hilt of a sword that had been concealed by his side. They did not dare to move.....
...I did not know, he answered. 'But I am Aragorn, and the verses go with the name.' He drew out his sword, and they saw that the blade was indeed broken a foot below the hilt. 'Not much use is it Sam?' said Strider. ' But the time is near when it shall be forged anew.'
In both the books and the film a lot hangs on Argorn's willingness/ability to take on his destiny as the King of Gondor/Arnor. In the books this gradually emerges over time, hinging on his decision to use the Palantir to directly challenge Sauron shortly after the destruction of Orthanc. In the films this is shifted to more of a defined moment of decision just before he take the paths of the dead and being given Anduril by Elrond.
This change is probably more about cinematic pacing for the films and puts a point of drama in what is otherwise a bit of a slow passage in the story and gives more prominence to the symbolism of the sword and the relationship between Aragorn and Arwen, which although important in the original story is somewhat overshadowed by other events and only really spelled out in the appendices.
As an aside it is also worth mentioning that the scene in the PJ films where Narsil is reforged (which I think only appears in the extended edition) bears very little resemblance to real world swordsmithing.
