catedrático
This appears to be a professor, but not a generic word for professor. From the definitions I’ve read, this refers to a special type of category of professors — those who have reached the highest status at a university (but this can also refer to those at a high school). The gist I get from this definition that I got from the Diccionario de la Real Academia Española (DRAE) is that a “catedrático” is akin to at least a full-time professor, not an associate professor or an adjunct professor, and quite possibly this title is reserved for those who have tenure. This notion seems to correspond with information I found later on a Wikipedia page titled, “Academic ranks in Spain.”. It is part of a larger page titled, “List of academic ranks.” It contains links for 75 other countries in addition to Spain; 7 of these 75 countries are Spanish-speaking countries — Argentina, Colombia, Costa Rica, Cuba, Mexico, Peru, and Venezuela). The only one among these that I see that refers to a “catedrático” is Costa Rica as in “Profesor Catedrático” and it refers to a “titular professor” (something below an emeritus professor and an “associated* professor). I do not know if any of the 14 Spanish-speaking countries not referenced by this page use “catedrático” among their terms for academic rank. (If you happen to know and/or want to set up a page for the academic ranks of your country, I am sure Wikipedia would encourage it and many others would appreciate it.)
*I think this may be a typo for “associate” professor.
docente
Appears to be more often used as an adjective than a noun. In fact, the DRAE only lists this as an adjective for teaching, but despite that, you will see use of “docente” refer to a teacher even in Spain’s paper of record — El País (see below).
Examples:
«… un docente de inglés es como un cocinero …» —Clarin
«Ésta es la propuesta para la evaluación de la actividad docente de los profesores universitarios diseñada por la …»—El País
«… un docente con más de 15 años de experiencia tiene un salario …» —El País
regional notes for «docente»:
After a review of academic rankings by country (as found on Wikipedia), only a couple of countries (Colombia, Costa Rica) appeared to make special mention of “docente.”
In Colombia, there is a distinction between a “profesor universitario” and a “docente universitario.” A quick glance at the Wikipedia page, “Academic ranks in Colombia” seems to indicate that becoming a “professor” in this country may be difficult and competitive, but worthwhile, whereas a “docente” is, perhaps, a position one takes on the way to becoming a full-fledged “professor” and can take the form of an “associate” or “adjunct” professor (to provide something of a U.S. equivalent).
In Costa Rica, you will find a “Docente de Emergencia.” I did not find any resource to back this up, but it seems likely that this would be the equivalent of a “substitute teacher” in the U.S. The word “docente” also applies to auxiliary teachers and lab assistants in Costa Rica. In the U.S., I think you would refer to these types of teachers as “Teacher Assistants/Helpers.”
maestro
From the Latin root “magis,” meaning “more” or “great.” (Source: Wiktionary entry for “maestro.”) As I understand it, in Spanish, this is just a generic word for "teacher," but can also mean “master.” It is a word that is actually found in several different languages, all with slightly different meanings, but at the core of them all is a meaning of someone who has mastered something.
Examples:
«El maestro ideal, según la Reforma Educativa» —Educación Futura
«Un "maestro del fuego" reveló los secretos del asado en La Rural» —Clarin
profesor
I’ve kind of already defined “profesor” when I attempted to give a description for “catedrático” and “docente.” Like “maestro,” this is a word that can be used somewhat generically (but perhaps a bit less so) and in certain contexts, it can have a very specific meaning. Generally speaking, however, this tends to refer to a teaching position at a university. It can also refer to a teaching position at a high school, but I sense the word is mainly reserved for those who teach at an institution beyond “la escuela secundaria.” A recent glance at a Univision headline has me second-guessing myself on this, so if any native Spanish speakers know for sure, please comment.
Examples:
«Profesores en huelga: ‘Mañana no vayan al colegio porque …’» —Perú21
«Un profesor de la Universidad de Navarra escribirá una película ...» —20minutos.es
While doing this research, I stumbled upon two other synonyms for “teacher,” so I decided to add them to this post. They are below:
enseñante
Only listed as an adjective in the DRAE.
Examples:
«He conocido a enseñantes muy buenos» —La Voz de Asturias
«…el rol enseñante del docente…» —La Voz
educador
This is usually translated into English as “educator.” It is a word that is more formal and less generic than “teacher.” According to Wiktionary, it defines someone who is “distinguished for his/her educational work.”
Example:
«Justo Arosemena, el educador»
—La Estrella
Lastly, allow me to share with you two more things that might illuminate these words a bit more. The first is an image of the Google Ngram I ran for these words:

And lastly, is this article here, which really was the catalyst that took me from I don’t really know the difference between these words to maybe I should do some research and share what I learn. The article provides what I think is more of a philosophical approach to the differences between some of these words rather than the dictionary meaning/common application I tried to lay out, and because of that, you just might enjoy it more than anything I've written above.
«Docente, profesor, educador y maestro» —El Heraldo