What's so cool about a purebred, and how do I "make" one?

A purebred has parents that are all descended from pets of the same theme. For example:

https://www.sylestia.com/view/lineage/?pet_id=3842263

All the blue stars signify that his ancestors are all purebred Black Velvet Petunia lups. If you go back far enough, you'll see the orange stars of the original themes.

As for "making" them, you can breed them yourself from themes/purebreds. You can also buy them from other people in advanced search.