Tragic irony is synonymous with dramatic irony, which occurs when we the audience know more than the character knows—when we have privileged information. This happens immediately in Macbeth, when we meet the witches twice before Macbeth does, and each time we see them we know they are up to no good. Indeed, the second time we see them one tells her sisters she has been “killing swine” (1.3.2), surely foreboding ill for Macbeth. As for your second question, how dramatic / tragic irony affects the relationship of Macbeth to his wife, or what does tragic irony have to do with her influencing him, we see Lady Macbeth reading her husband’s letter and hear her reaction to it. She says to herself that he is “too full o’th’ milk of human kindness” (1.5. 17), which a warrior such as Macbeth would hardly like to hear his wife say about him, and then a few lines later she adds “Hie thee hither / That I may pour my spirits in thine ear, / And chastise with the valor of my tongue” (25-26), which would again insult his masculinity. In short, Macbeth does not realize his wife’s view of him, which he would find very offensive and derisive if he knew about it, causing him to be less likely to be persuaded by her if. I can imagine that if he had overheard her he would have said “how dare you speak about me that way” rather than “yes, dear, I will consider killing the king.”
Some other tragic ironies are also found in Act I. When Lady Macbeth, in support of her husband and his career, convinces him to kill Duncan, she is ironically ending her marriage to Macbeth. The irony isn't seen until later in the play, but her actions in Act I set up the situation that ends their marriage. Once Macbeth gives in to the temptations of the witches, he loses sight of everything, including his marriage. The Macbeths are a loving couple in Act I. By the end, Macbeth doesn't care when the doctor tells him his wife is very ill.
The other irony with Lady Macbeth also stems from her convincing Macbeth to kill Duncan. She is the hard-hearted wife in Act I who is trying to help her husband further his career. Yet, it is Lady Macbeth who later feels the pangs of guilt so deeply that she can't function, and it leads to her death.
Another irony comes from Macbeth's honor by King Duncan at the beginnning of the play. Bestowing on him the title of Thane of Cawdor would normally represent the actions of a heroic character, but Macbeth proves himself to be anything but heroic. An ironic tragedy is also tied to the crimes Macbeth committs. The more he sins through the commitment of his crimes, the less freedom he has. With each crime, Macbeth loses options in what he can do to protect himself and his reign as king. By the end, his options run out.