I don't know if you have to memorize these terms for your studies, but the idea of "quasi-static" process is somewhat obsolete and misleading. The division between "reversible" and "irreversible", on the other hand, has a clear mathematical definition, involvig a time-reversal operation (which consists in replacing $t$ by $-t$ and making some so-called "parity" transformations, which depend on the system under study).
Basically the point is that to describe a thermodynamic system you need to choose some constitutive (or closure) equations. These join the universal conservation equations of the theory (mass, energy, momentum, rotational momentum, etc.) to give a system of equations of motions that can be solved. The ideal-gas equation relating pressure, density, temperature is an example. Another example is the relation between stress and velocity for Newtonian fluids. Yet another example is the relation between heat flux and temperature gradient in Fourier's law of conduction.
Each such equation is only applicable to specific systems in specific regimes. It's not universal. That's why it's called a "constitutive" equation. Note that the same system can be described by different constitutive equations when it's in different regimes.
Some constitutive equations allow for non-reversible processes, others allow only for reversible processes.
Now imagine you have a thermodynamic system and you want to mathematically describe its behaviour within a particular regime. Which constitutive equations will you use?
It turns out that for some systems, if the change in their dynamical variables (which include temperature and density) is slow, then we can use constitutive equations that yield reversible processes. That's why such regimes are called "quasi-static".
But slow with respect to what? It depends on the physical constants of the system. Some regimes can be considered "quasi-static" even if the changes are quite fast from a human point of view (I believe that some explosion processes, for example, can be considered as "quasi-static"). So that term is misleading.
There's an analogous situation in relativity theory: when should you use the equations of Newtonian mechanics, and when those of relativity? If the velocities involved are enough small compared to $c$, then you can use Newtonian formulae. You could call such situations "quasi-static". Of course they can involve amazing velocities of the order of kilometres/second – which are small compared to c.
There's another reason why "quasi-static" is a misleading term. Some regimes of fast change can be described by constitutive equations that yield reversible processes, and if you slow them down, they cannot be described this way anymore and you must treat them as irreversible!
To summarize: The distinction reversible/irreversible is first of all mathematical; it concerns constitutive equations and the motions (processes) you can obtain from them. The distinction quasistatic/non-quasistatic is blurry; it concerns whether you can apply specific mathematical descriptions to some experimental situations (with the caveat that "quasistatic" may actually mean "very fast" in some experimental situations). Usually you can avoid using the latter distinction altogether, and simply specify which constitutive equations you're using and which process you obtained from them.
Neither distinction is set on stone. Some situations can be described by reversible prcesses in some cases, and by irreverible ones in other cases, depending on the purpose of your investigations. (Just like some motions can be described by Newtonian mechanics or by relativity theory, depending on your goals, precision needed, etc.)
Some good references (from very different fields) about these points, if you're interested:
Astarita: Thermodynamics: An Advanced Textbook for Chemical Engineers (Springer 1990) [A truly wonderful book!].
Owen: A First Course in the Mathematical Foundations of Thermodynamics (Springer 1984).
Samohýl, Pekar: The Thermodynamics of Linear Fluids and Fluid Mixtures (Springer 2014) [I warmly recommend chapter 2 of this book].
Truesdell (ed.): Rational Thermodynamics (Springer 1984).