Generally speaking, APRS data in the RF environment is very short-lived.
Once gateways between local RF APRS activity and the internet APRS-IS were invented, interest arose in storing APRS data for later review. By storing the APRS packet data in a database things like tracking become more interesting.
Any internet user can do a search of the database to look at station position reports, messages, any data that made it's way into the APRS-IS system.
Probably the best known APRS database is findu.com. A more complete list of useful database sites is below.
This aspect of APRS has become overwhelmingly popular. People set up their station PATH so that their packets reach an I-Gate and make it into the findu database. This use has in many ways overshadowed the original purpose of APRS as a local, tactical, RF-based tool.