I've found SQLite to be adequate for my purposes on Unix/Linux boxes, since it supports most of the important SQL stuff, but not all the more complex clever stuff that MySQL, PostgreSQL etc do. It's also tiny in comparison. I don't generally need anything too clever in the way of databases, so smaller and easier suits me fine.
...but as others have said, this is only the backend, you'll quite possibly need something pretty to allow you to do stuff. Since my databases are all automagically talked to by PHP and Perl code, I don't need a general front-end. I believe that there are some GUI front-ends for SQLite, but I've never tried them, so can't advise.
(SQLite also runs on Windows, MacOS (!), etc).
