After the death of Pain, the fake Madara fights Konan to know the location of Pain's body...During this he mentions that the Rinnegan is his and he wants it back...I couldn't understand this..
Was Pain born with Rinnegan or was it implanted by fake Madara?
Nagato was not born with the Rinnegan. It was implanted into him by the real Uchiha Madara before his (Madara's) death.
Uchiha Madara activated his Rinnegan at an old age, so he made a plan by which he could be revived using the Rinnegan's Rinne Tensei jutsu after Akatsuki captured all the Bijuu into the Gedo Mazo. He could then proceed with the Moon's Eye Plan.
He implanted his Rinnegan into Nagato, without Nagato realising it, and then made Uchiha Obito assume his identity as the fake Madara, so that Obito could both collect the Bijuus using the Akatsuki, and control Nagato to use the Rinne Tensei to revive Madara.
Source: Naruto Manga Chapter 606. I don't want to paste screenshots here, since the entire story is described fully in that single chapter.
No, he wasn't born with the Rinnegan. According to the wiki(emphasis mine),
When Nagato was a child, Madara Uchiha transplanted his own eyes into the young boy without his knowledge
One day, during the Second Shinobi World War, two Konoha shinobi attempted to find some rest and food in their thought-to-be-abandoned home. Believing that the shinobi were there to kill them, his parents died during the ensuing chaos in their effort to protect Nagato, as confused and shocked, the Konoha ninja mistook them for enemy shinobi. After witnessing his parents death, Nagato in his grief used the Rinnegan for the first time and killed the assailants. Nagato would eventually come to regard this as the first of two great sources of pain in his life, only later realising that it was him who had killed the two shinobi.
