PostgreSQL (http://www.postgresql.org/, see also http://wiki.postgresql.org/)
- BSD license ( ultra-free )
- the pgAdmin (gui) is nice enough; psql, the commandline client, is great.
- *very* active & helpful community; reported bugs often fixed within hours. There is often instant help at #postgresql / freenode.
- SQL:2011 ( see: SQL Conformance )
- ultrareliable: yes
- DBD::Pg (see: DBD::Pg )
Postgres is supported on a wide range of platforms (see the distributed buildfarm status page (which continually tests if compiling is still faultless) - not even all platforms are represented)
SQLite is only good for 1-person use, and then only with limited data size (the SQLite website honestly admits this). It is therefore IMHO not really a serious candidate for the epithet 'database', but YMMV, and you can see from the other replies that opinions vary wildly and widely. SQLite /is/ good for embedded use (which is a usage that postgres does offer).
update: Added mild SQLite bashing & some wordsmithing.
update (2012.10.27): s/SQL:2008/SQL:2011/g;