There are multiple options to learn languages in Madrid. 

First of all, there are Official Schools of Languages, which are public institutions. There is one in each district, and also there are some in the main Comunidad de Madrid municipalities.

Then, it is possible to study in the cultural institutions that are from the country of the language, such as British Council or the American International Institute for the English language, Goethe Institut for German, Institut Français for French, or Istituto di Cultura Italiana for Italian (there are also institutions to learn Chinese, Russian, Persian, etc).

Last, there are a good number of private academies in which it is possible to learn different languages.