Loki's mother is Farbauti, a giantess who appears in
Thor & Loki: Blood Brothers. So he's a full-blood giant, but he came out a runt, which is why he's roughly the size of a god rather than a giant. Giants, much like everyone else, have a range of sizes and appearances, too. Some are barely taller than a typical human or god, others are the size of skyscrapers; some are blue, some are caucasian, some look like they're made of ice.
His father's name is Laufey, so his full name is Loki Laufeyson. Fun fact: in the actual myths, his parents' names are reversed, so he'd actually be Loki Farbautison there. I'm guessing Marvel switched them because Stan Lee loves that double-initials thing so much (or maybe he just didn't know any better, given that the Norse myths are sort of confusing to read).
Odin did not ever lay a hand on Loki's mother because he's not at all Loki's father. Laufey was a king of the frost giants who treated his runt son like utter s***. Laufey's giants went to war with the gods and the gods won, with Odin himself felling Laufey in battle. (Or so they thought; JMS' run features a flashback that shows Laufey was mortally wounded by Odin but actually finished off by a time-traveling adult Loki. Comics!
) Odin found Loki after that and, since he couldn't bring himself to murder a child (and also because there was a prophecy, which was secretly that same time-traveling Loki manipulating Odin, again according to JMS' run), decided to raise the boy as his own. There were never any plans to unite the giants and the gods in the comics, unlike the movie. Also unlike the movie, Loki was adopted when he was a young boy, not an infant, so everybody always knew about his true parentage. Thor accepted him regardless, but no one else did. Sif hated him instinctively, sensing that he had some darkness in him, Balder just ignored him as much as he possibly could out of sheer indifference, and everyone else only respected him because he was their prince, adopted or otherwise. Comic Loki's descent into villainy is actually way more justified for two big reasons: 1) everyone knew he was a frost giant and treated him like an outsider (except for Thor) his whole life, and 2) Odin was nowhere near as noble in the comics as he is in the movie, so he treated Loki like utter s*** a lot of the time.
Knowledge, motherf***ers: now you has it.