Although I think I understand the basic definition, I don't see why we need the concept of a quasistatic process? Which equations or concepts in thermodynamics actually rely on this kind of process?
'Quasistatic process' means that the system studied (like gas in a cylinder) undergoes changes so slow that at any time, thermodynamic variables like pressure, temperature, internal energy and entropy have definite value and provide meaningful description of the system. Consequently, equations like
dU = TdS - pdV
are valid for such processes. That's why the concept of quasistatic process is useful.
The fact that the process is quasistatic and the above equations are valid do not guarrantee that the process is thermodynamically reversible. Whether it is depends not merely on what happens to the system, but also what happends to other systems that it interacts with.
For example, consider the system studied is at temperature $T_1$ and is giving off heat to another, colder system at $T_2$ through poorly conducting heat bridge.
------------------- heat ----> -------------------
| system 1 at T_1 |===================================| system 2 at T_2 |
------------------- poorly conducting rod -------------------
The process of heat exchange will be quasistatic, the above equation for entropy change will apply to both systems individually:
dS_1 = dQ_1/T_1
dS_2 = dQ_2/T_2
while $dQ_2 > 0$ and $dQ_1=-dQ_2$.
But the process is fundamentally irreversible, as heat is flowing from warmer to colder body and entropy in the end will increase.
So quasistaticity only guarrantees applicability of thermodynamic quantities and relations between them; it does not by itself guarrantee the process is reversible.