Tapas, in the sense of morsels of food you get for free with your drink are called 'aperitivos' in Madrid. You’ll most often find the aperitivos in the smaller, more neighborhood bars but some bars of this sort still remain in old downtown Madrid. The fancier tapas bars SELL tapas and raciones (larger portions for sharing). Free tapas are usually something small like a plate of olives, potato chips, slices of chorizo, slice of tortilla de patata... but also a bit of stew or a coffee-saucer of paella on mid Sunday. Half the fun is exploring and looking for your own treasure trove of free food.

Etiquette in Madrid:

Aperitivos come with a glass of beer, wine, vermouth from the tap or 'mosto' (no-alcoholic grapes juice), often with coke or soda, but never with mineral water, coffee, brandy or spirits.
Aperitivos are for people at the counter, not for tables.
Customers can't demand or choose the aperitivo, is the bartender choice.
The aperitivo will be different with the second drink. It is common to get bigger or better tapas for each consecutive drink that you buy (up to a point), but this is by no means a fixed rule.
The aperitivo for two people is not necessary a double-sized aperitivo served to a single.
Do NOT order tapas or raciones BEFORE your aperitivo has come because you probably will lose your aperitivo for free.
Aperitivos are served all the week long but the best time for good aperitivos is from 1pm to 3pm when the cook is at work.
Free tapas map: