Another possible scenario is if you need to create some sort of DBA GUI along the lines of pgAdmin, either because you have specialist requirements, just as a programming exercise, or because you think you can do better!
However, the main purpose of this article is as a lead-in to further articles on CRUD which has obvious and widespread uses.
The article provides a simple introduction to the subject, as well as creating the database schema which I'll use for future articles.
To answer your question "Would it be as an alternative to using SQLite as a library?", PostgreSQL is an industrial strength RDBMS which can hold its own against SQL Server and Oracle in a client/server environment so you wouldn't use it for standalone applications.