Well, it's Saturday night here and I just popped open a beer, but here's what happens (and it's happening all of the time). This is the bare bones account. If you crave more detail, maybe somebody will fill that in, or you can check this out , now that you'll have the basic idea, in a Biology text:
DNA makes mRNA (transcription). One side of a DNA strand is the template. A-U, C-G, G-C, T-A for the DNA to mRNA respectively. The mRNA is then processed. Some segments are cut out by enzymes. These are the introns. The remaining segments, the exons, are spliced together. This is the mRNA that will be translated into an amino acid sequence by a ribosome or ribosomes.
I've read different ideas, speculation I think mostly, about why this processing occurs. One idea I read is that the DNA that transcribes the introns may represent viral DNA that got spliced into the host cell genome. Some of the Molecular Biology types here could probably give you some info on that.