For me, the term "Quasi-bound States" is frequently recognized when talking about quantum wells. Sometimes, it is referred to "Quasi States" as you mentioned.
A quasi state is generated when a quantum well or double heterostructure is formed. For example, a 10-nm-thick GaAs is sandwiched by double AlGaAs barriers. From this, GaAs/AlGaAs quantum well is obtained. Let us consider the conduction band, due to the finite thickness of GaAs layer and the surrounding barriers, electron states in GaAs become quantized in a form of energy band (not infinitely narrow!).
From application point of view, the concept of quasi states can be applied to "resonant-tunneling diodes" where the carrier transport and thus the conductivity can be controlled by external bias. You can see more details in Chapter 4 of Physical Models of Semiconductor Quantum Devices by Ying Fu.
The above discussions are very good for understanding and i wish to add one more point. If you look at the Fermi energy in QWs we used to call that as Quasi-Fermi energy. This can tell you only about the number of carriers in that state; not more than that. So in order to differentiate from the conventional terms we used to add the scientific prefix "Quasi".