Arganzuela

Is in the South of the city.  It is an old, traditionally working class area with very reasonable rents, but not very attractive from the touristy point of view (Metro Delicias or Legazpi)

Barrio del Pilar

(Metro Barrio del Pilar) is a residential area, not highly attractive but with reasonable rents and one of Madrid's few shopping malls/centers "La Vaguada."

Centro

Could be attractive for students who want to live their Madrid experience to the fullest, given that here you can find a lot of cultural activities, night life and tourist attractions, along with an enormous number of restaurants.  A large number of classical buildings are currently being refurbished, adding to the flavor of the old quarter.  (Metro Sol or Callao).

Chamartin

A well communicated area, (Metro Alfonso XII, Prosperidad and Republica Argentina) with prices that range from medium to high.

Chamberi

Another typical student area (Metro Alonso Cano, Bilbao, Iglesia).  It is a very central area and well communicated, with lots of old apartments, most of which have been renovated recently.

Chueca

Has become a focal point for the gay community, which has given new life to what was a very run down part of the city.  Note, however, that some areas that form part of the Centro are best avoided, like Lavapies and La Latina, as they have a reputation for being unsafe, as do some of the streets leading directly off of Gran Via.

Cuatro Caminos / Tetuan

(Metro Cuatro Caminos or Tetuan) is cheap and is not far from el Parque del Retiro, as well as being well communicated.  However, being a relatively inexpensive part of the city, it is home to a large number of illigal immigrants, and can be a conflictive area, particularly at night.

Fuencarral

To the North of Madrid (Metro Fuencarral and Herrera Oria) is not central, but the prices are not terribly high, and it is a safe area.  Not far away from Barrio del Pilar.

Hortaleza

A little far from the Parque del Retiro (Metro Arturo Soria and Mar de Cristal) but it could be a good area for students with families who are looking for a residential area - apartments here usually have garages and swimming pools.  Prices are reasonable, but not cheap.  Calle Arturo Soria tends to be somewhat more expensive.

Majadahonda, Pozuelo, Aravaca & Las Rosas

Another idea is to go to Madrid's suburbs, where flats tend to be newer and less expensive.  It is a good option for students who bring their families with them, given that apartments are larger with garages, gardens and swimming pools at quite reasionable prices.  These are some examples of this kind of area.  They are connected by local trains which link up to the metro at Principe Pio, Atocha, Chamartin, and Nuevos Ministerios, among others.

Mirasierra

Farther North, is more expensive and you could use a bus, but would be better off using a car.  It is a very nice residential area.

Moncloa

One of the areas favored by students in the capital, and a large number of university and postgraduate students live and socialize there.  The area is also very well communicated thanks to the Moncloa bus terminal.

Prosperidad

(Metro Avda de America, Alfonso XIII, Cartagena) is near Barrio Salamanca and is quite reasonably priced, except for new buildings which tend to be more expensive. 

Retiro

(Metro Retiro or O'Donnel) is another area that is very popular among students.  It is a pleasant and near the Parque del Retiro.  It is expensive, particularly the main streets, although it is possible to find reasonable rents in large, refurbished apartments.

Salamanca

Possibly one of the best areas in Madrid due to its central location and close proximity to the Parque del Retiro (Metro Avda de America, Goya, Serrano), but it is also one of the most expensive areas of the city.