The former Spanish national railway operator is RENFE. As most former monopolies, these companies tend to be useless at communication (in the past they never had to use it, if you wanted to ride the train you were shoved "their" services. Period). So it's quite normal to be confused about acronyms they use all the time...
RENFE = train company operating, among other, the high-speed network in Spain. Their website is www.renfe.com
AVE, Alvia, Euromed, Talgo, Altaria... marketing names (products, so to speak) for high-speed/long-distance train services offered by Renfe. The difference between products is related to whether they are direct, semi-direct, with trains of this or that type, etc. In practice, to most effects, you shouldn't care about it: price, availability and schedule are all you want to know. As per the trains in itself, very often we're talking about the same (or very similar) model of train irrespectively of the product name
MD, Regional and Regional Express are trains that cover medium distance services (ie. inside an autonomous community, sort of your "states"). These train might have between few and fewer stops from origin to destination, so in practice, one could say they're relatively "direct". These trains are also served by Renfe.
Cercanías are trains that cover short distances, a.k.a. commuting trains and are mostly used to serve main cities and their neighbouring counties. Sometimes though, cercanías can also serve longer journeys, much like the above Regional services, but stopping at every station, thus much more slowly. These trains are also served by Renfe.
ADIF is a spin-off of RENFE that was created a few years ago in order to comply with the non-monopolistic regulations issued at EU level. This company is in charge of the network and stations while RENFE is in charge of the train service-only. In practice though, it's the same crap, with the same delay tactics, misinformation and horrendous service when there's a problem in the track.
To make it more interesting.... in certain zones (ie. Catalonia or Asturias among other), there are other train companies, which existed much before RENFE was even created in 1941 as a result of the forced nationalisation of the railway service imposed by the dictator Franco's administration. Thus for example, in Catalonia, there's a second train company, called FGC (Ferrocarrils de la Generalitat de Catalunya) that cover different lines than those served by Renfe -ie. going to Montserrat from Barcelona. Their service is far superior in terms of reliability and service... unfortunately though one can't really choose because they serve, as mentioned, completely different (physical) lines. Furthermore, for political reasons long to explain here, part of the Renfe service in Catalonia has been ceded to the Catalan government which is implementing changes to try to minimise the horrendous service provided by Renfe and, for example, the commuting service here is called "Rodalies" instead of "Cercanías". Also the information website for commuting trains, as well as MD, Regional and Regional Express within Catalonia is served by a less cluttered and easy to use website: http://rodalies.gencat.cat/en (or www.rodalies.tk for a shorter version)
Lastly to mention that, despite the above, train services are very popular here for their price -cheaper than in many other European countries. Also note that train travelling is perfectly safe.
Btw, tickets for MD, Regional, Regional Express and commuting services cannot be purchased online but on the station on the day of departure. (PS. Recently some changes have been introduced for a reduced number of MD trains, but best not to go there otherwise you'll end up even more confused!)
Confusing enough? :))))