There are two scenes in which Macbeth receives predictions from the witches. In act 1, both he and Banquo receive predictions, and in act 4, he is shown three apparitions that give further information and warnings to him.
In act 1, scene 3, Macbeth and Banquo are walking from a battlefield on the heath and come across three witches. The witches hail Macbeth Thane of Glamis, Thane of Cawdor, and King (of Scotland). At this point in the play, Macbeth is already Thane of Glamis, but he is surprised to hear the other two prophesied titles. He will learn later in the scene that King Duncan has, in fact, named him Thane of Cawdor after the former Thane of Cawdor betrayed the crown and fought with Norway. This makes Macbeth believe that the witches' predictions are correct and that he will be king someday.
He decides to take matters into his own hands, of course, and starts plotting to kill Duncan, which he does off-stage in act 2. In act 1, Banquo also learns that he will not be a king but he will "get kings," meaning he will be the father of kings. This is important later because Macbeth's paranoia leads him to have Banquo and his son, Fleance, murdered—however, Fleance actually escapes.
Once Macbeth has been king for a little while, and has committed several more murders to keep his power, he demands more information of the witches. At the beginning of act 4, the witches show him three apparitions. The first tells him to "Beware Macduff" (IV.i.71). Macbeth admits that he already is worried about Macduff, and later in act 4, he sends murderers to kill Macduff's family. Next, the second apparition says that Macbeth cannot be killed by any man "of woman born" (IV.i.80). The third apparition says he will remain king "until / Great Birnam Wood to high Dunsinane Hill / Shall come against him" (IV.i.92-94). The second and third apparitions seem to state impossibilities, so Macbeth's confidence increases after this meeting with the witches.
He does not adequately prepare for the coming war with England that will eventually depose him from power. All of these predictions do come true in act 5, as well. Macduff is the character who kills Macbeth, and it turns out that he was cut from his mother's body because she died before he could be born naturally. As the English army approaches Macbeth's castle on Dunsinane Hill, they disguise themselves with foliage, so it appears that the forest is actually approaching the castle.