Dependency Injection is a design pattern that allows an object A to declare its dependencies to an external object B that supplies those dependencies. The dependencies declared by A are usually class interfaces and the dependencies B provides are concrete implementations for those interfaces.
This allows for loose coupling of code because object A no longer needs to be concerned with initializing its own dependencies. Object B decides which implementations to provide to object A based on a configuration or desired behavior.
This is an important concept to understand for extension developers because it forms the basis of how Magento composes its classes.