An occurrence of a database component. This can mean very different things depending on the platform.
DB2 - An instance is a logical database manager environment where you catalog databases and set configuration parameters. Depending on your needs, you can create more than one instance on the same physical server providing a unique database server environment for each instance.
Oracle - A set of memory structures that manage data files. Every database has at least one instance. An instance can exist without a database. In RAC multiple instances share the same database. Database here is used in the Oracle sense of the word.
SQL Server - The software, memory structures and data files. A computer can have more than one instance. One instance can be designated the default, but secondary instances must be named. Some things are shared between instances.
If you are familiar with what instance means on other databases, please add.