The following is meant as a supplement to piq's excellent writeup on Dream Interpretation, not as a contradiction.
Dreams make sense. They even make the same kind of sense as 'real' life, or at least our mental interpretations of real life. They are our interpretations of real life.
You can understand dreams in the same way as you understand real life, simply by following the linkages of thought and association between the elements of the dream. There is very little fixity of symbol and meaning between people, which means that 'dream dictionaries', while a useful way of sparking your own thought processes about the dream, cannot provide you with a reductive interpretation of it. It has been established by Carl Jung and others that we all share a collective unconscious, populated by recurring archetypes which are shared by all people, even from totally different cultures - however, you cannot fully interpret the content of your dream through archetypal analysis any more than you can know a city from studying a map.
There is no need to build up a vast store of symbols and their common meanings, as if learning a new language. The language of dreams is your language. It's a very personal thing. Just as you know your own thoughts, you can know your own dreams. Jung had an enormous knowledge of symbols and stories, but never forgot to find out from his patients exactly how they felt during each part of their dream.
How did the dream feel? What does it bring to mind? If you are wondering what a certain event in the dream means, ask yourself, how did it make me feel? How did I react? What else makes me react that way? If there is a character in your dream about whom you are wondering, ask yourself, how did I feel about them? Who in my life do I feel that way about? If something terrible happens, ask yourself, what am I afraid of that would feel like this?
Not only is this way of interpreting dreams easy and powerful, but it also has the side effect of integration - this kind of enquiry integrates your conscious thought processes with your subconscious dreams and thinking. Rather than distancing yourself from your dream as if it were a theoretical entity, a movie you saw last night, you are living it, remembering it and thinking about it as if it actually happened to you. It did actually happen to you, and is as available and comprehensible as your everyday experience in the waking world.