Winston's dreams are indicative of his inner, even repressed, thoughts and feelings. They represent parts of his character that he may not want to show others, or even himself.
First, Winston dreams of his mother, whom he has not seen since he was a child. His last memory was her calling after him as he ran away with his sister's chocolate ration. He never saw her again, but dreams about her. Unfortunately, his dream is not a happy on as rats end up crawling all over her body. This dream symbolizes Winston's guilt at losing his family, his guilt at his own selfishness. It also foreshadows his future torture in the Ministry of Love.
Secondly, Winston dreams of the Golden Country. The Golden Country is an imaginary place associated with hope, the hope of the rebellion's success and the hope of spending his life with Julia. Here, the land is bright, beautiful and sunny - a contrast to the dark, dingy environment in which all citizens life. This symbol of hope shows Winston's belief in success even in the face of obvious adversity.
In 1984, Winston has a number of dreams that are significant because of their symbolic value. In Part One, Chapter Three, for instance, Winston dreams of a place called the Golden Country in which he sees the dark-haired girl (Julia) strip naked while running through a field. This place not only represents Winston's repressed sexual urges but also his desire to experience true freedom, portrayed here by Julia's removal of her clothes. When Winston wakes up from this dream, he has the word "Shakespeare" on his lips. This is also significant because it emphasizes Winston's desire to return to a time before the Party took power. Just like the Golden Country, Shakespeare is reminiscent of the liberty and freedom of the past.
Similarly, in Part Two, Chapter Seven, Winston dreams (again) of his mother. This time, however, he realizes that he was not responsible for his mother's death; she just disappeared after Winston stole his sister's chocolate ration. Winston's vindication is important here because it leads him to a deeper conclusion about the Party. Specifically, that the Party is so powerful because it has eradicated private loyalties--the type of loyalties that were central to the lives of people, like Winston's mother. This powerful realization spurs on Winston in his rebellion against the Party.