I have slightly modified the following from a Yahoo! Answers question I answered sometime ago.
He is referred to as a god in the Nibelungenlied and the Völsungasaga. In the Eddas he is generally referred to as an Ás, i.e., one of the gods called the Æsir. An example of this is in the Gylfaginning, from the Prose Edda, in which appears also the most explicit reference to anything Jötunn [Giant] about him. He and his two brothers Býleipt (or Býleist) and Helblindi are therein said to be the sons of a Jötunn named Fárbauti, while their mother is a certain Laufey, who is also called Nál. We are not told much about Laufey beyond the idea that the reason she was called Nál, "Needle," is because she was "slender and weak".
It is possible that Loki has similar origins to most of the other Æsir, who, like Óðinn and Þórr (Thor), are at least half-Jötunn, since the mothers of these gods were giantesses. Because Loki was among the treaty-makers who commingled their blood to become sworn brethren when the gods were making this pact, he became a member of this group, all of whom were considered to be Æsir from then on, whatever their nature or origins, as Vanir (a tribe/race of deities which for a time were the Æsir's rivals) or Jötnar [plural of Jötunn].
Loki's shadowy origins also make it difficult to ascertain whether he was initially pure Jötunn or not. We know nothing about his brothers apart from their names, so they do not help clear this issue up. Their parents are a bit of a puzzle themselves. In a late document called Sörla þáttr eða Heðins saga ok Högna ("The Tale of Sörli, or of Heðin and Högni"), the Jötunn Fárbauti is described as though he is merely a man, and an old one at that, at the time that he sires Loki. Elsewhere he is said to have impregnated Laufey with Loki by striking her with a bolt of lightning. Could this be a euphemism for a sexual assault? And were Býleipt and Helblindi conceived in the same way?
It is sometimes said that Loki was a cousin of Óðinn (Odin) and that this was the reason that Óðinn had to accept him among the Æsir. If this is the case then it means that one of Loki's parents could have been a sibling of Óðinn's mother Bestla, herself one of the frost giants. At any rate, John Lindow has an interesting insight on the issue of Loki's nature in his Norse Mythology: A Guide to the Gods, Heroes, Rituals and Beliefs, in which he says the following in his article on Loki's mother Laufey:
What is most striking about Laufey is that we always read of Loki
Laufeyjarson and never of Loki Fárbautason ... There were no last
names in old Scandinavia ... One had a given name and a patronymic,
except in those rare cases when the father was unknown or unsavory, in
which case one had a matronymic. The fact that his father Fárbauti was
a giant was presumably something that Loki—and Odin—would rather not
be reminded of, especially since in this mythology kinship is reckoned
exclusively through male lines. (Consider the fact that Odin has a
giant mother and sex with giantesses is one of his weapons.) Was
Laufey, then, a goddess? She is listed with goddesses in one of the
thulur, and having a goddess mother might have been what enabled Loki
to be "enumerated among the æsir," as Snorri puts it in Gylfaginning.
If Laufey was a goddess, then Loki's genealogy as offspring of a giant
father and a goddess mother would be the same as that of his children
with Angrboda, namely the Midgard serpent, Fenrir the wolf, and Hel,
all great enemies of the gods, and this might explain his ultimate