The relationship between Leon and Mathilda is a rather complicated one, and is pretty much the cornerstone of the whole movie.
Leon is hinted to be mentally challenged in some capacity. He can't read and despite his prowess as an assassin is very naive and is easily fooled by Tony (and Mathilda). He also has problems with intimacy and letting people get close to him. His only real friend before Mathilda was his plant. At first he struggled a lot with allowing Mathilda to enter his life, going as far as attempting to kill her while she slept. Soon enough, Mathilda becomes his friend and a surrogate daughter figure.
Mathilda on the other hand is smart and savvy beyond her years. She comes from a dysfunctional home where she is despised by everyone. She too has no real friends. She's go starved for human contact that Leon's passing attempt at small talk immediately makes her latch on to him and offer to get him his groceries. Eventually, as her relationship with Leon progresses, she finds herself falling in love with him. Leon (and the audience) are left to ponder if it's true attraction or just an infatuation with the only adult-figure who has been kind to her.
The two broken human beings flock together and form a family-unit of sort. To an audience used to normal families, a very young girl with a crush on a mentally and emotionally challenged hitman is very uncomfortable. In fact, American test audiences were so uncomfortable that a scene where Mathilda (unsuccessfully) propositions Leon to sleep with her was cut for the American release. One could argue that this discomfort is an intentional part of the movie.