Depends on your project... SQLite works great for getting a database up and running quickly without a hassle of an install. If you're planning on getting into enterprise-level stuff, Oracle has a free version of their database you can run on your PC as well as Microsoft SQL Server.
Here are some of the DBMS's that can be installed on a desktop computer at no cost:
1. Apache Derby
2. Oracle MySQL/MariaDB (the open-source fork of MySQL)
4. Oracle Community Edition
5. IBM DB2 Community Edition
6. Microsoft SQL Server (free edition)
7. SQLite (used internally in Linux and Android OS)
Those are the most popular, but there are others as well. MariaDB and PostgreSQL are provided in standard package form in almost all Linux OS distros in addition to being available directly from their respective development websites for other platforms.
Apache Derby is now included as part of the Java JDK, which is a well-kept secret, since I myself only found out about this recently. I've been using a downloaded independent copy of it for years.
SQLite is the SQL DBMS built into Android, but it's also popular as a default or internal database on a lot of Linux apps. My Gourmet Recipe Manager app uses SQLite. So don't be surprised to find it was pre-installed on your Linux computer when it was first set up.
The Community Edition versions of the commercial DBMS's these days are fairly complete implementations of their commercial counterparts. The main difference is the number of CPU cores and/or amount of RAM that they'll be permitted to use and of course, you get less in the way of direct vendor support. Then again, direct vendor support these days is usually pretty feeble regardless of product. Used to be IBM would keep an engineer on-site in many cases. Then came the telephone waits. Finally things deteriorated to message boards staffed by junior personnel reading scripts and possessing questionable English skills. I'm picking on IBM since Service was their primary selling point up until about 1990, but their competitors are no better. Often the open-source products have better message boards these days with the added bonus that if you get frustrated enough, you can always dig through the source code (and I have, alas, more than once).
"privilege" comes from the Latin words for "private" and "law" (legal) and dates to feudal times. To "claim privilege" meant that you were above the laws that applied to the common people.