Elena
Elena, the girl who more or less healed our hero and brought him out of his 5 century state of depression. With her help, Stefan grew to become more comfortable with himself (enough to accept his nature and remain a vampire in book 4) and was able to love other people again. He loved her completely, it was pretty much as simple as that. There wasn't anything he wouldn't do for her, and was quite at a loss when she died. For all the things she helped him come to terms with and all the ways they helped each other grow, I still can't find enough words to describe their connection. It's love. Who can really capture that? (for more depth on their relationship, read his love. )

Katherine
While it's difficult to fully explore the nature of Stefan's early relationship with Katherine since we're only given flashes of what their life together was like, I'll do my best. Katherine had come with her father to stay with Stefan's family in Florence for the summer to help her recover from her previous illness. Drawn by her grace and innocence, Stefan fell in love with and formed a tie with her so powerful that effects of their relationship were still being felt centuries later. She was his first love, and it's almost certain that the two would have remained companion forever had the competition with Damon not presented itself. His relationship with Katherine was the turning point in his life, because she brought bitterness, sorrow and cynicism to his life where it had not been before. His feelings for her eventually dissolved when he fell in love with Elena and Katherine's deception was exposed, but it's clear from scenes in The Dark Reunion, that he harbored no ill will towards her.

Friends

Bonnie
Stefan's relationship with Bonnie is a friendly one, but I never got the impression until the last book that he thought of her as anything more than a friend of Elena's. Sure he cared about her well-being, but it wasn't a close attachment. As we know from reading The Dark Reunion, he hadn't given Bonnie his phone number or address in Florence, so there obviously wasn't much intent for either of them to keep in touch. However, I think the bond was strengthened somewhat in the last novel as they had to work even closer together.

Meredith
As I already said above with Bonnie, I don't think there was much more to their relationship outside of their mutual connection to Elena. They may have liked each other all right, but I don't think there would have been much interaction between them if Elena wasn't somehow involved. I think that's a real shame, though, since I think Meredith and Stefan have enough in common to found a lasting friendship. Perhaps had there been a longer gap between his second arrival in Fell's Church and Elena's resurrection, the two could have found something within each other. But since there wasn't, I'm sure their relationship will always be loose, and revolve around their individual connection to Elena.

Matt
Though this relationship suffered strain at times because of Elena or Stefan's nature, Matt was Stefan's only true friend (excluding Elena) in the series. There are many scenes that exhibit the spirit of camaraderie that passed between them. Through Matt, Stefan found the sort of belonging that he hadn't experienced since his mortal life. We don't see enough of Matt's point of view to understand his bond to Stefan, but it's clear that he considered him a real friend since he did stick by him under circumstances where even the best of people might run. But then, Matt was just a saint.

Family

Damon
A strained relationship from the beginning. Stefan tells Elena in The Awakening that he couldn't remember when Damon started hating him, but that he thought that the resentment was tied to the early death of their mother. Other causes for conflict seem to be in their different relationships with their father; where Stefan was respectful, Damon was defiant and his actions shocked and upset the younger brother. I've always thought that Damon had much more animosity towards Stefan than vice versa, as is often found in younger-older sibling relationships. Meaning that Stefan's feelings towards his brother were a reaction to his brother's ill feelings more than they had any grounding of their own. After the conflict with Katherine, resentment deepened to hatred, fear and guilt that would last until the end of The Fury. In The Dark Reunion, their relationship is a lot better than it had been, but issues of trust and their mutual exasperation for each other's life styles and points of view keep the brothers' relationship tense.

Giuseppe (Father)
Not much is known about Stefan's relationship with his father, but Stefan seems to have respect for Giuseppe (as was proper ^^) and did call him a "fair" man. He wasn't mentioned much, so I can only assume that he wasn't too terribly close to his father, but that the relationship was probably fairly good and normal for the children of his time.