What happens to Lady Macduff and her child in Act IV?
To put it bluntly, all are brutally killed by Macbeth's henchmen. You see, once Macbeth obtains the crown, he becomes more and more obsessed with holding on to power. As a result, he vows to eliminate anyone who he feels is a threat to him. With that in mind, he knows that Macduff has gone into hiding, and Macbeth is afraid Macduff in plotting against him. In an effort to draw Macduff back to Scotland (he is in England trying to convince Duncan's son Malcolm to join him in a rebellion against Macbeth -- he really is plotting against him), Macbeth dispatches assassins to kill Macduff's family.
The killing of Macduff's family is one of the more bloody acts in the play. Lady Macduff begins the scene complaining that her husband has gone away and left her all alone -- she even suggests she might leave him and find another man to replace him. Her son says he sees through her complaints and points out she still really does love Macduff. After this exchange, Maceth's men swoop down upon them and kill them both -- first Lady Macduff and then her son (after she told him to run away).
So, by this point in the play, all of Macduff's family is dead -- murder by his kinsman turned enemy Macbeth.
In Act IV, Scene II of Macbeth, a number of henchmen arrive at the Macduff castle in Fife with orders to kill Lady Macduff and her son. (Remember that in the closing lines of Act IV, Scene I, Macbeth tells Lennox of his plans to kill the family, which he expresses as "give to the edge o' the sword.")
In this scene, Macbeth's henchmen enquire after Macduff's whereabouts and pronounce him a traitor before turning first on Lady Macduff's son. This is shown clearly though the stage directions in which the young boy is "stabbed" repeatedly. On witnessing this violence, his mother flees the castle, shouting "murder," but the henchmen give chase and there can be no doubt that she is quickly captured and killed.
These acts of extreme violence show that Macbeth will go to any lengths to protect his crown. In the next scene, however, Macduff is galvanized when he learns of his family's fate and swears that he will have revenge.