Estimated reading time: 5 minutes.
No TL;DR, but here is a short answer to your question:
Definitions were arrived at after looking at a number of sources to include Collins, Tureng, WordReference, and Reverso. If you're looking for lots of detail, I highly recommend taking a look at the Collins entries for all four of these words. To get you started, here's the Collins entry for "romper." I'll explain the other columns briefly.
The "Frequency" column indicates how often the word is found and the lower the number, the more frequent the word is. The frequency list I used came from www.opensubtitles.org. The column to the right of "Frequency" is pretty self-explanatory. The source used for the "Etymology" column was Wiktionary. Collocations came from linguatools. Numbers in parentheses represent the number of hits for each collocation. This column shows you how interchangeable these words can be (especially "romper" and "quebrar"), but also how much more frequent "romper" is.
If you can't see the chart very well, open this page up in Chrome, then simply right click on the chart, and open it up in another tab.
The answers in this thread are good, but you'll find even more information about these words (especially "romper" and "quebrar") on the web. Apart from the "short answer" I gave above, the rest of this answer will just focus on "romper" and "quebrar," simply because that's what led me to this thread in the first place and I feel that others have sufficiently addressed "quebrantar" and "partir" already. Below are some excerpts from discussion threads I thought were pretty good regarding "romper" and "quebrar" and, at the same time, offered something new or supported something already mentioned in this thread. A link to the full thread from which each excerpt came follows each set of excerpts:
The above advice is helpful, but if you don't mind me going off on a slight tangent, keep in mind that "quebrar" can also mean to go bankrupt. "Romper" is never used to mean to go bankrupt.
—StrangerCoug; New Member; El Paso, Texas
So...In case "You have broken my heart" use quebrar or romper?
—noncasper; Senior Member; China
Not exactly; quebrar is used when something "long" is broken into two or more pieces, for instance a bone of the arm/leg or a stick, but neither a plate or the heart.
Romper is to stop working, but also to break, including those situations where you may use quebrar. So, in case of doubt, use romper, and there will be no problem.
—Duometri; Senior Member; Madrid, Spain
Quebrar para mí implica que haya grietas o fisuras, quiebras un hueso porque se agrieta y entonces se quiebra, pero es una percepción subjetiva mía. Romper es mucho más general, no implica necesariamente partir algo.
Para un corazón, definitivamente "romper": Me has roto el corazon (you broke my heart).
—nanel; Senior Member; Madrid, Spain
Sorry Nanel, bur I must contradict You.
I think, in this context, "quebrar" have a more widespread use.
De la prensa: "a primera vista, no parece un joven apuesto capaz de quebrar el corazón del género femenino"; y: "me quebró el corazón a escuchar las noticias del terremoto en Perú."
—Eclisse; New Member; Italy
—From "romper - quebrar," a WordReference discussion thread
Using "quebrar" instead of "romper" with "el corazón" appears to be just a personal preference, but may be more frequently seen in books than in online news articles. Across all mediums, "romper el corazón" is far more common, but according to what I found in CORDE and CREA, as well as this Ngram here, it appears that "quebrar el corazón" was the only phrase used to express this sentiment until about the 20th century. Only in recent times (i.e., the last 20 years or so) has "romper el corazón" been more frequently used than "quebrar el corazón," which could indicate that someone who prefers the latter version of this phrase may be from an older generation.
I´m just looking at my dictionary here (Collins). The examples for quebrar include bones, health, heart, mood (depression), body (twist or bend at the waist).
The much longer list for "romper" includes windows, curtains, paper, dishes, furniture, machinery, rope, clothing, balance, silence, contracts, relationships ... but also bones.
—From "'romper' versus 'quebrar,'" a Span¡shD!ct discussion thread
My textbook uses romper for body parts (to break a bone) but my friend told me that native speakers never use romperse for body parts and that fracturarse should be used or possibly quebrarse and that romperse is for objects like a piece of paper. Is this so?
—KitKat913; New Member, United States
Then your friend is not from Spain, for sure, because here we use "romperse" all the time to refer to broken body parts (me he roto la nariz, se le ha roto la muñeca, etc...). "Fracturar" is fine to describe broken bones, but it sounds a little too technical, and usually only medical staff use it.
—BillBasque; Senior Member; Basque Country, Spain
—From "romper / fracturar / quebrarse," a WordReference discussion thread