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I've been looking around, and I haven't found any good explanations on what each of the sockets from the Texture Coordinate Node does. Can somebody please explain this?

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Each socket represent a different mapping function that maps surface coordinates to the texture coordinates. What you get from the socket is 3D vector array which can be manipulated with vector or math nodes to alter the mapping (usually using the Mapping node).

The different mapping options are explained here:

How do texture coordinates work?

How the vectors itself and math works and what you can do with it is explained here:

How do I create repeating patterns with cycles' procedural textures

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