<p>What is JDBC and where can I start learning about?</p>

<p>I know it's a way to access databases with Java, but what problems does it solve? Is it an ORM (or does it try to be)? Does it abstract away differences between databases, at the syntax level? What does it do? and what does it not do?</p>

No, JDBC isn't an ORM. It's the
Java Database Connectivity
API and basically it provides a database-agnostic access layer with a provider model (so that new database drivers can be added easily). Vendors can add more functionality for specific database features if they wish, but developers can ignore those features if they wish to work with multiple databases.
There's no mapping involved - just modeling for connections (and pools), prepared statements, stored procedures, result sets etc.