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Library
A library in context of game programming with LÖVE is a collection of useful functionality that make your life as game developer easier.
For example, a library may contain routines to simplify the handling of game objects such as monsters or powerups, or it could provide functions to handle sound more easily.
The scope of different libraries vary greatly. There are libraries that provide a single function to implement object orientation within Lua, to libraries that implement full blown game engines. Naturally, libraries with a narrower scope are more easily to integrate into your already existing project, while bigger ones impose a certain programming style. This is not necessarily bad, as setting a frame will often result in faster progress once you learned the concepts behind such a library.
Not every library is suited for your game and programming style. You have to pick and choose which one fits best.