What are dreams? - Some of the posts above have good explanations.
Do they mean anything? - I've learned from a psychologist that the meaning, or rather, the purpose of dreams is to keep you asleep. For example, if the blanket falls off of you during your sleep, you'll dream about the North Pole or something that explains the cold. This also explains how something that happened in a dream happened in real life. For example, if you somehow hurt yourself during your sleep (But not strong enough to wake you up), you might dream that someone or something hit you in the hurt place. People often wake up with bruises or small cuts, and think it happened because they were dreaming it. It's actually the other way around, they were dreaming it because it was happening during their sleep.
Is there any way to make yourself have a dream? - Every human actually dreams every night, but forgets their dream a couple of seconds after waking up. The best way to remember a dream is to wake up while dreaming it. That's why we remember most of our nightmares - the fear wakes us up while we're dreaming it.
Also, here's an interesting fact about nightmares: There are several parts of your brain that are active during "normal" dreaming. During nightmares, however, they are less active. The more active brain part during nightmares is the part responsible for memories (The hippocampus). It's as if you're actually remembering those scary events, instead of dreaming them.