What is PostgreSQL? How important is it? It's it necessary? What does it do?

Also with the trial version, and the registered version, what are the differences? And if I do register/buy the full version, is it a one time fee or is it recurring?

PostgreSQL is a database server. Specifically it's the type of database that PT3 uses to store all of its data. You need access to a PostgreSQL server in order for PT3 to function properly. The download that comes with PostgreSQL has the PostgreSQL installer so that you can install PostgreSQL on your system. It only needs to be installed once, so that's why we have versions that don't include that installer for people who are upgrading. If you want to find out more about PostgreSQL, you can go to their website and they have extensive documentation.

The trial version lasts for just 60 days, and there are three things you cannot do:
(1) You cannot create custom statistics.
(2) You cannot use multiple databases.
(3) You cannot use TableTracker.

The fee for PT3 is a one time fee and it gives you a free one month trial of TableTracker. TableTracker is optional and has a monthly fee if you wish to continue using it beyond that point.