Update June 8, 2013
Bright yellow airport express buses
The new airport express (express aeropuerto) bus connecting Madrid Barajas airport and the city centre 24 hours a day came into service last week. The bus leaves every 20 minutes during the day, and after 11pm at 25-35 minute intervals. You can take the bus from outside the arrivals (lowest floor) at T1, T2 and T4 at the airport, where it will take you directly to the centre, stopping at O'Donnell, Plaza de Cibeles and Atocha Renfe railway station. However returning to the airport you can only take it from Atocha Renfe bus interchange as it makes no stops. After midnight the bus terminates at and leaves from Plaza de Cibeles (the stop is outside the massive post office building where Paseo del Prado meets Cibeles). The journey takes 30 minutes and costs 5€. The buses are easy to spot as they are painted bright yellow. For more details and the bus timetable, see the Exprés Aeropuerto page on the EMT website.
At the airport there are other city buses (101) to take you to Avenida de America. The 200 bus will take you to Príncipe de Vergara 102 and you can go from there. A 10 ride metrobus ticket can be used on these. You can find these at tabaconists and the newspaper stands. These tickets are not for individual use and can be shared. If you don´t have a an "abono", the cost is 1,50€. There is a metro stop "Avenida de America" which will take you north as opposed to the center of Madrid. If you are flying into Terminal 1, the bus stop is right there! Very easy!
This is a great bus to take back to the airport, as well. Goes every 15 minutes!
Continuing with the metro
You can also connect with metro line 6 from the O' Donnell stop. The stop at Cibeles leaves you next to Banco de España metro for line 2, and from Atocha Renfe railway station, you can connect to line 1 of the metro- you need to enter the station, go down the escalators and follow signs to the metro.
The bonobus can be used on both metro and bus. You can buy these at the metro station, which is very useful!!!