Referring to a group of organisms which are grouped together genetically because of their common ancestor, but which do not include all the descendants of that ancestor.
For example, birds and mammals are descended from reptiles, but are not normally classified as reptiles.
In the taxonomy usually used, it is legitimate to have a taxon called Reptilia which includes all ancestral reptiles, and all present-day "reptiles" as commonly understood, but not their cousin taxa Mammalia or Aves.
In the stricter evolution-based classification of cladistics, this is invalid. The only taxon they recognize is the clade, which must consist precisely of a common ancestor and all its descendants: so we are reptiles in that sense.
Other examples of paraphyletic taxa are Dinosauria, if birds are descended from dinosaurs; and Amphibia because reptiles (in both senses) are descended from amphibians; and invertebrates, because they do not include their descendants the vertebrates.
The opposite, a taxon that does include all its progeny, is called monophyletic.
A third term is polyphyletic, for a grouping which has no genetic basis, coming from more than one ancestor. Some Linnean classifications were polyphyletic, based on functional or structural similarities, but which DNA evidence now shows to be from convergent evolution. As polyphyletic groups are discovered they are reclassified, because it is no longer considered a valid basis of classification.
An asterisk notation is sometimes used so that the conventional groups can be named even within a strict cladistic classification: so if Reptilia is monophyletic, including mammals and birds, the notation Reptilia* is used for the paraphyletic grade traditionally called reptiles.