Research or other work done after getting a PhD is postdoctoral. Your cousin must be pretty happy about the postdoctoral fellowship she got at the University of Hawaii. Surf's up!
When you stay in school long enough to earn a doctorate degree (usually called a PhD or Doctor of Philosophy), you can describe the work, writing, and research you do as doctoral. It follows that related work done after that — but generally before getting a full-time academic job — is postdoctoral, since post- means "after." A postdoctoral student, for example, might work for a few years and teach some classes before pursuing a university teaching position.