A trigger is the Oracle database equivalent of a rule. A rule is effectively something which is triggered as a result of a specified database event, such as inserting into one table triggers a rule which will insert another record into another table. Triggers, rules and events are generally used to enforce referential integrity and business rules.
Referential integrity can be enforced using rules which can be used to force reference matching between primary and foreign keys between related tables. However, most databases provide more simplistic methods of enforcing referential integrity.
Business rules can be constructed using rules and events. In Oracle a trigger will simply send a message somewhere. This trigger message could be a simple log message or an instruction to tell the database to perform some other action, such as a table insert on another table. A set of rules can be added to a database which could make the database a knowledge base or even an expert system.