"Wish" is to hope for something. To "desire" something means you have a sense of longing for something or someone. "Dream" is something that happens when you sleep...unless it's a daydream, which is when you think about lovely things happening (you act out mental visions of scenarios).

To add a couple of points already made: To wish for something is to hope for something that might be unlikely to happen, will not happen, or seems almost impossible for it to happen. In addition to the process that occurs while sleeping, "dream" can also refer to a goal they would like to achieve. We say it is one's "dream" because it's a vision one has within their mind of how it will unfold. "It is my dream one day to climb the highest mountain in the world." In an example such as this, the person usually can see the goal of reaching the top of the mountain within their mind.