Laguna's parents are not mentioned in the game, but he does have a wife, an adopted neice, and a son. The latter being slightly important to Final Fantasy VIII. Now, like the dutiful webmistress I am, I shall analyze.
Though never stated outright, it is widely accepted that Laguna is Squall's father. The evidence is enormous. Ward mentions that Squall looks like his mother, not like his father, so thus Squall's parents must be someone Ward knew. When Squall encounters the Moombas, they call him Laguna, because the Moombas have some sort of blood-recognition sense, and Squall's blood reminds them of Laguna, for some reason. Laguna has Squall's card, implying that Squall is most important to Laguna. (Ma Dincht has Zell's card, General Caraway has Riona's...) Squall, Raine, and Laguna all have matching water names. Furthermore, there's the general RPG rule of thumb that if two characters could possibly be related, they are related.
The character design in Kingdom Hearts shows an older Squall with longer hair, presumably to look more like Laguna. (Much like Cloud's character design was changed so he looked more like Vincent, only there's actually a logic behind this one.)
The blood relationship between Laguna and Squall ties the game together. Laguna, like many other characters, serves as a foil for Squall. Whereas Squall is hardly the model hero, Laguna's arc provides a more traditional protagonist for the player to relate to. On this note, Laguna is not a secondary character in the main plot, he's the main character of a secondary plot. We see the story of Final Fantasy VIII develop from two decades and through two perspectives- that of Squall and Laguna.
As I said before, Laguna provides a foil for Squall- someone to highlight his personality by contrasting with it. Laguna is easygoing, funny, bumbling, and caring, where Squall is silent, brooding, serious, and has some serious relationship issues. Laguna has lost not one, but two loves, his world has been upturned time and time again, but he survives and moves on. By contrast, most of Squall's emotional problems come from his feeling of abandonment after Ellone left him.
For all their differences, there is one unmistakable trait that links them: they are both natural leaders, undeniably the heroes of their stories. Everything seems to fall into place for them, in spite of tragedies, and they always find themselves at the head of the party despite their obvious shortcomings. Though Laguna can't find his way out of a paper bag, and sometimes it seems Squall has the emotional IQ of a turnip, they are time and time again called to the forefront and succeed where no one else has. They have the unchallanged respect of their peers and the devotion of their subordinates. I'm not sure that there is a gene for heroism, but they've both got that.
If Laguna had known about Squall at the time of Raine's death, it's fairly certain he would have risked life and limb to get him in his custody. If they could have had a more traditonal father/son relationship, I think Squall they would have had a positive influence on each other, in much the same way that it is the outgoing Rinoa that forces Squall out of his shell. Of course, then the game wouldn't have happened, and there where would we be? As it stands, Squall may not be very happy to find out Laguna is his father, and indeed, may be angry with him for being basically absent the whole of his childhood. On the one hand, I find it difficult to believe that anyone could stay angry with Laguna for long, on the other, Squall does prove exceptional in the antisocial department. It'd be interesting to see what happens to these two after the end of the game, but there's a reason the end was put there in the first place. The reparation of the family Loire/Leonhart is a different story.
Ellone is arguably the catalyst of the entire game, provides the link between the story's past and present and accordingly has the special gift of time travel. She is an orphan who was adopted by Raine. When Laguna came into their life she made an instand bond, calling him uncle and playing matchmaker the way kids do.
It makes sense that Laguna would get along with children, in some ways he is just an overgrown child himself. His simple love for life is something that children readily understand, and so Laguna and Ellone found an unshakable love in each other. It is, after all, Ellone that calls Laguna away from his wife, and leads to his becoming President of Esthar. Unfortunately, in all the chaos of rebellion, Ellone winds up at the Matron's orphanage with Squall.
During her time in the orphanage, she and Squall become very close. Squall is reliant on Ellone, and when she is sent away for her own protection, Squall is devestated. Later on, she is rescued by an Estharan spaceship and she and Laguna are reunited. She probably explains the truth about Squall to him, as well as what happened to his wife. Ellone uses her time-travelling powers to reunite Laguna and Raine one last time.