When I talk about dreams on this site, and when oneirologists talk about dreams in their research, we are talking about the actual felt experience of the dream as it happens, and it refers to all internally-generated experiences (i.e. they come from our own minds) that happen during sleep. Dreams are different from dream reports, which we use in research.
Because there are four stages of sleep, which are very different from each other in many neurophysiological ways, the content and characteristics of dreams in different stages of sleep vary widely. For the types of dreams that are typical for the different stages of sleep, take a look at the pages for each stage of sleep. They are: Stage 1 non-REM sleep, Stage 2 non-REM sleep, Stage 3 non-REM sleep, and REM sleep.
There are lots of different ways we can categorise types of dreams, such as lucid dreams, recurrent dreams, typical dreams, hypnagogic and hypnopompic dreams bad dreams, and nightmares, just to name a few. We know that dreams picture things from our waking-lives, and the ways in which this happens are being researched, in terms of what things from waking life we dream about, and the temporal patterns that they follow such as day residue and dream lag.
There are many theories about why we dream, including cognitive theories such as emotion-processing and memory consolidation, and psychodynamic theories such as wish-fulfilmment and compensation; I will continue to add more of these to the glossary in the future. We can do all kinds of things with our dreams, and many people do, for example using various dreamwork and dream interpretation methods.