A Utopia is the "perfect" society, which does not exist and is in fact impossible. Think John Lennon's "Imagine." The reason a utopica is not possible is because there always has to be some kind of governing body ruling the society, and this would lead to oppression in some way and, thus, would not be perfect.
Dystopian societies are those that have tried to become utopias. They are often depicted in novels and movies. Anthem by Ayn Rand is the perfect example of a dystopian society. In trying to treat everyone as equal, the people lost their individuality and face severe consequences if they try to be different in any way or think for themselves. This book, and others like it, are written specifically to show what would happen if we tried to create a utopian society.
These words are most commonly used with regard to the societies they represent. Dystopian society is usually described with dark, dreary words that convey negative ideas as misery, oppression and squalor. Utopian societies are more the ideal of what a perfect society should be. It is interesting to consider that usually governments are responsible for attempting to control society, and it is possible that if a government considered a society utopian, the citizens might consider it dystopian.
Utopia and Dystopia refer to types of society. A Utopian society is something mankind strives for because it represents a harmonious condition of perfection and peace in all areas of society. There are not destructive conflicts between competing ideologies and such.
A Dystopian society is one where certain authorities impose what they deem the right society on people, even if a majority of the people disagree with them. They seek to control people under their authority. They may believe their dictates are for the betterment of society (as they see it), but a Dystopian society oppresses their people and does society harm, while maintaining they are doing what is good for the people.
To me, these both refer to societies in which people have tried to engineer things so as to achieve some sort of perfection. In a utopia, the people have tried and succeeded in creating a perfect society. In a dystopia, they have tried to create what they thought would be a perfect world, but have ended up creating a horrible world through their actions.