Background: Prehistoric dinosaur who survived (just like Nessie :))
Motivation: Has changed over time, from a creature that wanted to be left alone, to a violent and vicious animal, to an occasional saviour of Earth, to a largely territorial animal.
To be honest, many of these questions have been well explained (and referenced) on the Wikipedia page:
Within the context of the Japanese films, Godzilla's exact origins
vary, but it is generally depicted as an enormous, violent,
prehistoric sea monster awakened and empowered by nuclear radiation.
So it can be seen that whilst it's certainly empowered by nuclear radiation (and so was very topical upon its release), it wasn't created by that. In fact, his origin is much earlier:
In the late Cretaceous era (70 million years ago) there was a species
of therapod dinosaur resembling a large Tyrannosaurus, but with
special amphibious adaptations which allowed it to swim between the
islands upon which it hunted. This dinosaur species has been dubbed
"Gojirasaurus". A minimal breeding population of gojirasaurs somehow
survived the great extinction event which killed off other dinosaurs
and continued into the modern era...
This description tallies with the background of Godzilla in the 2014 movie, which you specifically referenced. This quote is from the advertising for an official toy for the film:
Possibly the last of an ancient species of giant amphibious creatures
that evolved when the surface of the Earth was over ten times more
radioactive than it is today. GODZILLA can create his radiation stores
into a violet, focused exahlation of atomic ray. Rarely seen, but
spoken of in ancient island mythers, "Goijara" was last spotted in
1954, when the U.S. Navy encountered him and attempted to kill him
with an atomic blast in the Pacific Ocean. Since then, the giant
creature has been living in the deep ocean - until a theat to his
survival from an ancient foe forces him to appear.
As to the main focus of your questions - his allegiance and motivations - I think that all the movies need to be examined to see how his "character" has developed and changed. Again from the Wiki:
Godzilla's allegiance and motivations have changed from film to film
to suit the needs of the story. Although Godzilla does not like
humans, it will fight alongside humanity against common threats.
However, it makes no special effort to protect human life or
property and will turn against its human allies on a whim. It is
not motivated to attack by predatory instinct: it doesn't eat
people, and instead sustains itself on radiation and an
When inquired if Godzilla was "good or bad",
producer Shogo Tomiyama likened it to a Shinto "God of Destruction"
which lacks moral agency and cannot be held to human standards of good
and evil. "He totally destroys everything and then there is a rebirth.
Something new and fresh can begin."
The above paragraph (with references removed) shows that Godzilla tends to follow no master but himself, acting as he sees fit. This is further discussed on the Godzilla-specific-Wiki:
Godzilla, in the original film...is an animal with semi-sapience that stumbles
upon human civilization without any malicious intent, only destroying
man-made structures or obstacles like buildings when the humans
Godzilla is a malicious entity created from the restless souls of the
dead from World War II. [Since then he has] developed as a character, and has
since become a savior of the Earth, saving the world from other
monsters like King Ghidorah, the Showa MechaGodzilla, etc.
According to Mothra's Shobijin's translation of Mothra, Rodan and
Godzilla's conversation in Ghidorah, the Three-Headed Monster,
Godzilla only "hates humans because they hate him."
In Legendary Pictures' Godzilla, Godzilla's behaviour seems to be that
of a territorial animal.
So effectively, Godzilla is perfectly capable of fighting and defending himself, and even rescuing humans, but largely wishes to protect himself and his territory.
Gareth Edwards, the director of the film, corroborated this idea in an interview where he was asked who "his" Godzilla would be if it was a person:
His answer: the last samurai.
"He's an ancient warrior who's the last of his kind, and his kind has
long since died out," Edwards says of his take on the legendary
creature. "He lives a very solitary lonely existence and he's very
happy to keep away from everyone, but we keep doing things to force
him to return and put things right." ...
To see Godzilla's motivation
and understand his reactions "when he was doing his thing," Edwards
adds, "we dialed some more personality and made him a lot more human
than we thought we would."
So you can see that the director's aim was to make him territorial, but ultimately "human" in nature - a guardian who wants to be alone, but keeps returning to protect us (a fairly common theme in many superhero movies nowadays as well).