What database aspects should be monitored?

Availability, performance, recoverability and capacity.

To be in more detail below are the items you can monitor very specific to database...
Windows / Unix Oracle Service present for DB Instance Windows/Unix TNS Service present for Listener Can successfully connect to Database Instance Suspended Sessions Exists Datafile cannot extent Autoextensible Datafile approaching defined maxsize Rollback Segments with >50 extents Processes approaching Processes Limit Sessions approaching Session Limit Recovery File Destination approaching defined maxsize No corrupt data blocks exists Blocking Sessions Exists Excessive Waits ASM Diskgroup usage approaching capacity Check Segments can extend Broken Public Database Links Exists Alert Log contains ORA- Errors, incomplete checkpoint messages or archiver error messages RMAN Resync has not run in last6 hours Data Guard Health Check No Data Guard Log shipping or applied gaps exists No failed database jobs exist Streams Health Check Database files in backup mode Flashback Health Checks Check Segments can extend Datafile and Redo Log Group limits Segments with PCT_INCREASE >0 Sessions Highwater Mark approaching Session Limit Invalid Objects exist Nologging Objects exist Statistics gathering failed or not gathered Excessive Redo Log Switches Orphaned Datapump Jobs Exists Passwords expiring in next7 days RMAN Backup log contains RMAN- errors ASM and Database Alert Log, and Listener log sizes approaching1 Gb Export Logs contains no errors Default Users not disabled