First let me start by saying that redox buffers are a made-up thing. All these QFM (actually FMQ) or NNO or other abbreviations don't exist in nature. They were developed as a way to "cheat" in experiments to get the desired Fe(II)/Fe(III) ratio, and later extended for more uses. High temperature and pressure experiments are permeable to hydrogen: it just goes through the experimental capsule. Depending on the direction of diffusion, it can either oxidise or reduce the contents of the capsule, changing the desired Fe(II)/Fe(III) ratio. Buffers "force" a certain oxygen fugacity. In reality, redox (or not) is buffered by a variety of reactions that we may not even know exist.
partial pressure (fugacity) of an element such as oxygen
In the deep Earth, there is no free O2 gas phase, so it doesn't make sense anymore to talk about the "partial pressure". But the oxygen fugacity controls the reactions involved electron transfer (i.e. buffers).
how oxygen fugacity influences mineral crystallization
In general, higher oxygen fugacity means there's a higher ratio of Fe3+ to Fe2+. This is particularly true in melts. So a melt with lots of Fe3+ will crystallise magnetite (for example) whereas a reduced melt with little Fe3+ will not. This is one of the differences between tholeiitic and calc-alkaline differentiation trends. Calc-alkaline melts are oxidised, so they crystallise magnetite early. Therefore, they never reach high Fe contents. Tholeiitic melts re reduced, and magnetite does not crystallise until very late, so it is possible for them to reach high Fe contents.
This is also important for sulfur. Will a volcano erupt SO2 or H2S? It all depends on the oxygen fugacity of the system.
I will also add that the bulk amount of Fe3+ and Fe2+ does not directly tell you where a rock is oxidised or not (with respect to oxygen fugacity). For example, a rock with 1% hematite, 98% enstatite and 1% magnetite will have more bulk Fe2+ than a rock with 50% magnetite, 49% quartz and 1% fayalite. But, the former is at the HM buffer while the latter is at the FMQ buffer (reduced).
I highly recommend reading this (preferably more than once):
Frost, B. R. (1991). "Introduction to oxygen fugacity and its petrologic importance." Reviews in Mineralogy and Geochemistry 25: 1-10.