What database aspects should be monitored?

One should implement a monitoring system to constantly monitor the following
aspects of a database. This can be achieved by writing custom scripts,
implementing Oracle's Enterprise Manager, or buying a third-party monitoring
product. If an alarm is triggered, the system should automatically notify the
DBA (e-mail, page, etc.) to take appropriate action.

Infrastructure availability:

• Is the database up and responding to requests
• Are the listeners up and responding to requests
• Are the Oracle Names and LDAP Servers up and responding to requests
• Are the Web Listeners up and responding to requests
• Etc.

Things that can cause service outages:

• Is the archive log destination filling up?
• Objects getting close to their max extents
• Tablespaces running low on free space/ Objects what would not be able to extend
• User and process limits reached
• Etc.

Things that can cause bad performance:

See question What tuning indicators can one use?.