In Spanish it’s enfermero (m) or enfermera (f). The job itself, the location where you go to get first aid (in a school or other public space) and often the sign on nurse’s offices, is enfermería. The root word is enfermo/a, sick person.
The Spanish educational system has suffered too many revolutions in the last 20 years; I got my degree (ChemE) in 1994 under a curriculum or “course plan” approved in 1954, the 1992 plan had already been approved, the next one is now in place, and a new one is expected in two years - therefore, right now the only thing that can be said for sure about “how to get a degree in X field in Spain” is “wait too long and the requirements will change”.
When I was in college, nursing was a “professional school”: for a long time, it had been the only 3-year degree in the country. When other 3-year degrees got put in place in 1992 (to give a degree to those who had previously been “mid-degree droupouts”), many nursing schools went up to 4, seeking equalization with those which had previously taken 5 years (licenciaturas) and differentiation from the former-dropouts. They achieved it, and now you can get a masters’ or a PhD in nursing.
There are other degrees such as ATS (Ayudante Técnico Sanitario, I’m not going to try and translate it) or ATS Assistant which have lower educational requirements (vocational track degrees rather than college), but most people wouldn’t be able to describe the differences in detail. The list of “who can be whose boss”, yes, but the differences can be complex: for example, only a doctor can diagnose and prescribe, but whether nurses can write “refill” prescriptions or not depends on the region or hospital’s requirements (when they do it, it’s considered to be “by delegation” of the doctor’s powers).
As for whether you’d be able to work in Spain:
the specific requirements for homologation of titles are managed by the Ministry of Education, you’d have to ask them (or the embassy in DC); there may be purposes for which you are already considered homologated (for example, I’m reasonably sure you’d be able to sign up for higher degrees with yours) and others for which you’re not (requirements for a permanent job with the UHC system are higher than for private jobs).