Danielle Faraday, originally born Danielle Widmore, spent the first 17 years of her life on an island in the South Pacific. She lived in a group of island natives that would later earn the name "The Others" - among these natives were her parents, Charles Widmore, the leader of the group, and Eloise Hawking.
They weren't the best parents ever. At a very, very young age, Danielle began showing an extraordinary ability to perform complex mathematical operations - Eloise jumped on this opportunity to slowly mold her daughter into a brilliant scientist, with the hopes that Dani would be the one to one day figure out the island and all its mysteries.
She met Benjamin Linus when she was nine years old, after Eloise asked Ben to babysit. Naturally, Eloise had an ulterior motive; Charles had a hunch that Dani and Ben would get along. The way he saw it, if his own daughter was hanging around Ben all the time, he'd have an extra pair of eyes keeping Ben in check.
They instantly became friends, just as Charles had hoped (...well, as friendly as a twenty three year old can be with a nine year old, and as friendly as Ben Linus can be with anyone), and Dani started following him around, learning about both the island and its inhabitants from him. He became something of a role model and mentor for her, which explains... a lot. Between her parents and Ben, she didn't really have any moral influences in her life.
In 1989, Charles sent Ben on a mission to kill a woman named Danielle Rousseau. Dani went with him. Ben didn't kill Rousseau - instead, he kidnapped Rousseau's baby, Alexandra. Dani helped Ben take care of her, for a little while. Charles didn't approve. In 1992, he orchestrated what was later known as the "Purge" - an island-wide killing of all the members of the DHARMA Initiative, a group that had began performing research on the island in the early 1970s.
Dani was the one who pressed the button that released the poisonous gas. As per her father's directions, she entered a code into the computer in a place known as The Tempest to DHARMA members; she had no idea what was actually happening until a year later, when Ben showed her the mass grave in the jungle.
Some time after the Purge, Ben had Charles exiled. That's when Dani's relationship with Ben fell to pieces, but she remained loyal to him - whatever he told her to do, she did, usually without question. She never killed for him, but she did other questionable things, like spying on other members of their group and ratting them out for violations of their code. (Ben was the one who usually killed them, later on, depending on the offense.)
Eloise decided, in 1994, to leave the island and take Danielle with her. After moving to England, Eloise instructed Dani to change her last name to something other than Widmore, for reasons Dani never quite figured out. She chose the name Faraday.
Shortly after, Charles contacted Danielle and gave her a research grant worth £1.5 million. With those funds, she set up a lab and began conducting research on time travel.