There are several different Socket APIs for Garry's Mod. They all have slightly different APIs, and for many of them it's unclear if they work at all in Garry's Mod 13.
LuaSocket is 'the original' socket API for Lua. It has a very simple API that closely matches the POSIX sockets API, plus some goodies thrown in (e.g. HTTP layer, MIME type handling).
While LuaSocket doesn't support Garry's Mod, it probably wouldn't be hard to create a Garry's Mod fork of it. This has the advantages of having a high-quality library and universally-recognised API, but it would involve maintenance of a binary module.