Graphical explanation
of basic phylogenetic terms

In each of the images below; the red circles are the species (or genera) included in the taxon, while the white ones are not included in the taxon. The pink bars serve to highlight the lineages present in the taxon.
 
Monophyletic taxon :  A group composed of a collection of organisms, including the most recent common ancestor of all those organisms and all the descendants of that most recent common ancestor.   A monophyletic taxon is also called a clade.

Examples :  Mammalia, Aves (birds), angiosperms, insects, etc.

Paraphyletic taxon :  A group composed of a collection of organisms, including the most recent common ancestor of all those organisms.   Unlike a monophyletic group, a paraphyletic taxon does not include all the descendants of the most recent common ancestor.

Examples :  Traditionally defined Dinosauria, fish, gymnosperms, invertebrates, protists, etc.

Polyphyletic taxon :  A group composed of a collection of organisms in which the most recent common ancestor of all the included organisms is not included, usually because the common ancestor lacks the characteristics of the group.

Polyphyletic taxa are considered "unnatural", and usually are reclassified once they are discovered to be polyphyletic.

Examples :  marine mammals, bipedal mammals, flying vertebrates, trees, algae, etc.