Why do eukaryotic organisms have introns in their DNA?
There is still a lot to be learned about the roles introns play in biological processes, but there are a couple of things that have been pretty well established.


Introns enable alternative splicing, which enables a single gene to encode multiple proteins that perform different functions under different conditions. For example, a signal the cell receives could cause an exon that is normally included to be skipped, or an intron that is normally spliced out to be left in for translation (the Wikipedia article on the subject has a basic overview of the possibilities). This would not be possible, or at least would be much more difficult, without the presence of introns.
In recent years, we have discovered that RNA molecules (especially small RNAs such as siRNAs and miRNAs) are much more involved in regulating gene expression than previously thought. Often the small regulatory RNAs are derived from spliced introns.


There is probably more, but essentially introns enable a finer level of regulatory control. Biological complexity is often not the result of having a larger complement of genes, but of having additional layers of regulation to turn genes on and off at the right times. Prokaryotic genes are often organized into operons, and a single polycistronic mRNA will often encode multiple proteins from multiple adjacent genes. Since the biological processes required to sustain microbial life are much less complicated than those required to sustain eukaryotic life, they can get away with much less regulatory control.