How to prevent deletion of the first row in table (PostgreSQL)?

The best way I see to accomplish this is by creating a delete trigger on this table. Basically, you'll have to write a stored procedure to make sure that this 'default' category will always exist, and then enforce it using a trigger ON DELETE event on this table. A good way to do this is create a per-row trigger that will guarantee that on DELETE events the 'default' category row will never be deleted.

Please check out PostgreSQL's documentation about triggers and stored procedures:

http://www.postgresql.org/docs/8.3/interactive/trigger-definition.html

http://www.postgresql.org/docs/8.3/interactive/plpgsql.html

There's also valuable examples in this wiki:

http://wiki.postgresql.org/wiki/A_Brief_Real-world_Trigger_Example

Keep in mind how triggers work. They will fire off for every row your delete statement will delete. This doesn't mean you shouldn't use triggers just keep this in mind and most importantly test your usage scenarios and make sure performance meets the requirements.

Should I use a rule or a trigger?

From the official docs: "For the things that can be implemented by both, which is best depends on the usage of the database. A trigger is fired for any affected row once. A rule manipulates the query or generates an additional query. So if many rows are affected in one statement, a rule issuing one extra command is likely to be faster than a trigger that is called for every single row and must execute its operations many times. However, the trigger approach is conceptually far simpler than the rule approach, and is easier for novices to get right."

See the docs for details.
http://www.postgresql.org/docs/8.3/interactive/rules-triggers.html