Do you want to start a new degree in Spain? Or just spend a year as a part of your uni program?
Admissions in Spain aren't that "elaborated". You get the grades, you get admission no matter what (interviews and so on are unthinkable, tbh). You'll have to get a kind of conversion of the grades you obtain (A-Levels probably?) to Título de Bachillerato and Selectividad (or something like that). Every year universities publish a list of all the subjects and the grades the person with lowest grades who got in, i.e., if you get that or more, you'll be accepted. Bear in mind that grades published are always the ones from last year, 'cos they adjust the minimum grade according to the number of applicants, etc.
As to how to do it for a foreigner, absolutely no idea. We, Spaniards, are registered with a university depending on the high school we attend. At that university, we do our final exams and then apply online (to that uni or any other, at least in Madrid) through the uni's website.
I wasn't quite interested in all this (as I was applying to British unis), but I can ask some friends if you can't get information from anywhere else.
Good unis depend on what your subject will be. For example, for engineering, "Universidad Politécnica de Madrid" is really high regarded; but it's one of the toughest. You'll probably be doing harder work than at Cambridge or Imperial, and yet fail some modules (almost nobody passes them all, they have to retake the following years). And I've been told by some friends that a lot of teachers are there because they passed some kind of exam and hence, they can teach awfully and nobody (but the students) will care. There are more modern universities (such as Carlos III, in Madrid, again) who even offer some degrees in English.
Hope this helps, OP!