(This page is really a dictionary entry. The word is much used on Wikipedia and probably not known by many readers, so I think we need this. Feel free to improve the definition, and/or come up with a systematic way to distinguish dictionary entries.)
In biological taxonomy, a grouping of organisms is said to be paraphyletic if they do not represent all the descendents of some common ancestor. Most schools of taxonomy advocate that groups reflect phylogeny instead, and so view the existence of paraphyletic groups in a classification as errors.
Many of the older classifications contain paraphyletic groups, especially the traditional 2-5 kingdom systems and the classic division of the vertebrates. This does not necessarily mean that older biologists meant to create them; more often it was just that they needed to have some taxonomy in order to organise the huge number of species in a way they could understand, and without modern scientific evidence, guesswork was required that later turned out to be wrong.
No doubt some of the currently fashionable taxonomies (as of 2001) will later turn out to contain paraphyletic groups, though hopefully as more evidence accumulates a system will develop that is satisfactory in the long term.