He was born to, supposedly, Laguna Loire and Raine Leonhart. We don't actually know if those are Squall's parents, but it's still implied. Raine died and, for whatever reason, Squall was put in an orphanage. He grew up there with a few kids, but I'll tell you their names later because it gets more important then. Later he's sent to this place called Balamb Garden and trained to become a special fighting mercenary called a SeeD. SeeDs goal is to oppose the Sorceresses, and all Sorceresses are very powerful and evil and have spooky magic. Skipping forward a bit he acquires his first GF (which stands for Guardian Force, not girlfriend). GFs are powerful, summonable monsters that you can harness from within yourself at the price of losing some of your memories as you do so. That's what happened to Squall, and thus, he forgot everything about his past except for a few small remaining feelings about the people around him, including trust issues because the person he was close with suddenly disappeared without a trace. That is the majority of why he is so cold and aloof. His instructor's name is Quistis Trepe, and she has a cult dedicated to her. Later he's put in a mission with Seifer Almasy, his rival, and some kid named Zell Dincht. Later he's put in another mission where he meets a girl named Selphie Tilmitt. Later he's put in a different mission where he meets a cute girl named Rinoa Heartilly. Remember her. Later he's put in some other mission and meets a useless idiot named Irvine. Irvine is a coward. Later after another mission (he does that a lot) he finds out that everyone other than Rinoa that he met was actually from the same orphanage as him and they all forgot about each other because of the GFs! Oh no.
TLDR, Squall was an orphan but he lost most of his memories, his mother is dead, his father is a moron, and his sister was separated from him which caused him to distance himself from everyone, but then he met a cute girl and became less emo.