It is clear that Tadashi looks up to Professor Callaghan a lot. The picture book version of the film describes Callaghan as Tadashi's favorite professor, and many fans assume that Tadashi was one of Callghan's top students. Having lost his parents at a young age, Tadashi may have seen Callaghan as a type of father figure--but whatever their relationship was, it is clear that Tadashi cared about him enough to risk (and lose) his life in an attempt to save him.
In the film, it is unknown if Callaghan truly cared for Tadashi, as he seemed unfazed to find out that his actions had caused Tadashi's death. However, some fans believed that he truly regretted what he did, especially after Abigail was rescued and Callaghan was sent to prison. The movie states that Callaghan was not always a bad man, so at one point, he probably cared for his students like any other teacher would, and Tadashi may have been particularly special because of his talent and intellect. Their relationship varies depending on fan interpretation--some see them as good friends, some see them as a typical teacher/student relationship, and others may see them as lovers.
Note: Please add only ones that are part of the RotBTD fandom