City areas in Spain are divided into "barrios" i.e. neighbourhoods. Compared to other cities, Madrid does not have that many "historical" things to see, but a lot of the city's flavour and local culture can be found in its various barrios. Each barrio is distinct with its own ambiance and crowd -- these are my favourite ones:
1) Malasaña: Located close to Plaza España, this barrio is probably the coolest one for youngsters to hang in. The streets are narrow and the houses have colourful, enviable balconies. Each street has a row of little bars and restaurants, from local "old man" ones to trendy hipster ones, to ones for "guiris" (us foreigners). Many venues here play jazz/rock concerts.Overall, the vibe here is alternative and cool, awesome to head out in for a night of tapas and cañas!
Closest Metro: Noviciado (red line) / Tribunal (dark blue line)
2) Lavapies: This is THE immigrant barrio of Madrid, and my favourite. It used to be the old Jewish quarter and today it's the home of most Bangladeshis, Moroccans, and Africans. It's probably one of the cheapest barrios to eat and drink at, and also to live. The vibe is quite hard core hippy. You will find many a dreadlocks, flip flops and ripped pyjama wearing people all around -- they are mostly budding musicians and actors who are trying to make it happen for them. Because the barrio is cheap and home to immigrants, unfortunatley it has many drug addicts and alcoholics roaming around -- something you will notice as soon as you get there. But not to worry, stay out of their way and they are harmless, and there are always many police in the area.
Closest metro: Lavapies (yellow line)
3) Anton Martin: Another very cool barrio, I would call it a mix of barrios Malasaña and Lavapies. Small winding streets, very trendy yet laid back little bars. This barrio is a maze, but really cool.
Closest metro: Anton Martin (blue line)
4) Huertas: Bang in the centre of the city, this is THE party barrio. Full of discos and clubs, it's where you'll always find a hoard of people heading out to paint the town red. The area must have atleast a 100 bars and clubs, all clustered in one area -- so it's a great place to hit if you want a big night out drinking, dancing and being silly. Many hostels are located here, and you will see many guiris, but it's your best bet for a night out.
Closest Metro: Sol (yellow/blue line)
A walk around these barrios is a must in Madrid!