Fox Mulder was born in 1961, the first child of William and Teena
Mulder. His sister Samantha was born four years later, and the two
enjoyed a normal, playful childhood, where brother and sister would
play baseball, board games, and occasionally tease each other, as
siblings do. In 1973, however, Fox's life changed forever, as his
beloved sister was mysteriously abducted from their house one night
while he was babysitting her. With, "No note, no phone calls,
no evidence of anything" ("Pilot" episode), Samantha disappeared from
the Mulders' lives, leaving Fox shocked, confused, and devastated at the family's loss. His life and his family changed forever. His parents
divorced soon after, their relationship having fallen apart after their
daughter vanished. Fox was left to grow up in a very strange situation,
still confused over his sister's disappearance, and having to deal with
his parents at odds with each other. It is most likely the
disappearance of Samantha which started Fox's obsession with strange phenomena,
as he never stopped looking for her, and always believed that one day,
he would open his eyes and she would be back in her bed at home, as if
she had never been gone.
Mulder moved away from the United States for university, and attended
Oxford in England, and went on to study at the FBI Academy in Virginia,
where he graduated at the top of his class. He was assigned to the
Violent Crimes section, and it was there where his talents were shown,
and he quickly became successful within the FBI. Because of this, he
had more leeway when it came to pursuing his own personal interests,
which is where he discovered The X-Files. What most people would has
considered a waste of time, and made up stories, Mulder was fascinated
with the cases, and, "read all the cases [he] could get [his] hands
on, hundreds of them. [He] read everything [he] could about paranormal
about the occult..." ("Pilot" episode). It was these cases which gave
him hope about the possibility of explanations which would be
considered by more as out of the ordinary, and also gave him hope about
finding more information about his sister's disappearance.
Throughout the series, Mulder is seen as obsessed with the X-Files, his devotion to which some can
even call religious. He readily accepts strange occurences, and, more
often than not, opts for explanations which have no basis of science
whatsoever. While his partner, Dana Scully, almost always looks for a scientific explanation, (which was normally the harder one to find), Mulder has no problem accepting answers which are not considered "normal". He even goes so far as to make an X-File about his sister and her disappearance, a file which remains active for most of the series. Mulder is so convinced that Samantha was abducted by aliens that he does not care what others think of him and his persistence, but keeps pushing to find out the truth that he so desperately seeks. His devotion to this, as stated above, could be considered religious. Agent Mulder has continuous signs of this devotion throughout the series, one being his famous "I Want to Believe" poster which hangs in his office. This poster, like Agent Scully's cross, is a constant reminder to viewers that each of the agents has a strong belief in something, and it shows in their own personal way.
While his partner is a Roman Catholic, Fox Mulder does not seem to be religious at all, but instead, rejects organized religion, and has been known to be scornful of it. Even so, when Scully is abducted in the second season, and Mulder finds her cross in the trunk of her abductor's car, Mulder takes it and carries it with him, and even at one point wears it. While he does not agree with Scully's religious faith, he still holds a certain respect for her belief in it, and the cross is a part of Scully which Mulder carries with him while she is gone.
Mulder's search for his sister Samantha was the driving force behind him investigating X-Files and looking for aliens. He was convinced that if he were able to prove the existence of extraterrestrials and UFO's, then "that would invariably confirm his sister's abduction" (Samantha Mulder Wiki). Throughout the course of the series, he meets various Samantha "clones", but never finds the real Samantha, pushing him to keep searching for her. He finds some answers along the way as to why she was abducted (her father used her as collateral in a deal with the aliens) and what happened to her during the remainder of her life (she was raised by a man Mulder despises, known as the Cigarette Smoking Man, underwent numerous alien tests, and apparently died in 1987, unable to bear the tests anymore). His persistent quest finally comes to an end, with his questions regarding his sister answered.
The following video is from the end of the episode, "Closure" in the seventh season, where Mulder finally finds closure on what happened to his sister, Samantha.