Libraries are software which extend the functionality of a programming language, usually by providing an API to complete a specific task. Different languages may have their own name for libraries, such as Perl modules, or Java packages. They also may also be implemented in different ways.
Dynamically-linked libraries are libraries that are contained in a file separate from an application's primary executable, and are loaded at run-time. This has the benefit of allowing multiple programs use of the same code both on disk and in memory, saving space. It also allows a program to load additional, optional or interchangeable portions of itself into memory at runtime.
On Windows, these files are known as, naturally enough, as Dynamically-Linked Libraries, from which their file extension DLL is derived.
On UNIX-derived systems, these files are known as Shared Objects, from which their file extension so is derived.
In implicit linking, an operating system sees that a program will require the use of a library, and loads it automatically. In explicit linking, the running program asks the operating system to load the library.
Statically-linked libraries are combined with a program's code at compile-time, their code and the program's code combined to form a single executable.
Some languages can create a library that when used as a library does one thing; but has the ability to be run directly, where it will do some extra task using, but not limited to, functionality that it makes available as a library.