Madrid districts are the administrative divisions that share Madrid in 21 areas. Usually, people stay in Centro (Sol, Gran Vía, Alcalá, …) but the Metro, takes you to anywhere.
It’s easier to separate pubs by districts. I’ll put a link to their facebook pages, a little description and timetables.
As you can see on the map Chamberí, Centro, Retiro, Salamanca, Arganzuela are the core where the tourism focuses. I’ll put a link in every district with a wiki about neighborhoods to orientate you guys 😉