OpenSource DataBase engine that supports most (if not all) of the ANSI SQL spec. Has support for things such as Triggers and Sub selects.
The client psql is not as good as the mysql client but there are apparently some good Web interfaces for it.
See 
http://www.postgresql.org for more info and PostgresVsMysql for a comparison with MySQL.
Postgresql under Debian by default uses IDENT authentication to authenticate a user connecting to the database. In english this means that you can only connect to the database with the same username as your unix login. This can be edited in /etc/postgres/pg_hba.conf (Host Based Authentication) by changing "ident sameuser" with "trust" to let anyone connect to the database with any username without a password, or "crypt" etc.. The file is well commented, refer to it for more details.
A couple of minor "gotchas". As above, check your pg_hba.conf file. Previously "password" authentication covered all forms of stored password. Now it is strict. If you are using md5 passwords, it MUST be md5. Also, in SQL-land, LIMIT used to be fairly forgiving about its arguments. It would accept MySQL format without blinking. Now it's strict - this caught me out on a basic weblog script I use which had been ported from MySQL. Finally, if you are a Debian user, don't believe the instructions in the update README! psql -e <dumpfile will NOT restore your database...you need to do psql template1 -e <dumpfile.
