Notions of "confined" and "metastable" states? What is the exact definition of terms "confined state" and "metastable state", in the context of quantum mechanics? 
Can we also have a "confined metastable state"?
Can we somehow easily link these terms to the character of the potential?
 A: There are no exact definitions in that particular corner of quantum mechanics; people will use those words (and other words like "squeezed") to mean a lot of different things in different contexts.
A wavefunction is generally held to be "confined" if it's more or less "stuck in a location for a human-observable amount of time." If you've got a 1D square well, for example, there are two classes of solution: ones that look like sinusoidal waves off at infinity ("free" states) and ones that attenuate exponentially as space goes on to infinity ("confined" states).
A state is metastable when you've got a shallow potential well connected to a deeper potential well with some "barrier energy" between them.
It is a feature of quantum mechanics called "tunneling" that a particle which is confined to a metastable state will eventually, as if by magic, find itself in a stable state with the associated energy dissipated into environmental degrees of freedom, even if it does not have enough energy to jump over the barrier "classically". In some sense it is like there is an energy-conserving quantum noise which offers short "kicks" to the particle which it immediately "takes back". These kicks will occasionally be enough to knock a particle over any barrier.
So, if you "confine" a particle to a "metastable state", such a particle is not really "confined" in the sense that it will not stay in the well as $t \rightarrow \infty$. But you could still speak of it and most physicists would understand that you mean that the barrier energy is being held high enough to prevent both classical transitions and quantum tunneling over the course of the experiment -- in which case, yes, you can be confined in a metastable state. 
You could also have "free states", which are needed to really get a meaningful "confined" state. A good example of how this happens is the sorts of Hamiltonians you see for molecular motors, which can have some $H(\theta) = A \cos \theta + B \cos (2\theta + \phi)$ behavior. There are states where the motor is just rotating mostly-freely, but there are also states where you are "bound" to one of two potential wells, one of which might be "lower" and the other of which might be "higher".
A: It is not so clear what you mean by confined. Maybe you mean "bound states"?
Anyway, about the state of a particle in a potential well, we can have bound states, scattering states, and "resonant states. (People speak also about anti-resonant states but I am not sure whether it is relevant to your question).
1) Bound states are first of all integrable in absolute square. Their intensity tends quite rapidly to zero as $r \to \infty$. In short, they are mostly confined to a small volume. A good example is the ground state of an atom. The electrons of an atom in the ground state remain forever in the atom if we don't disturb them by bombarding with particles, by placing in all sort of fields, etc.
Usually, in a given potential well, the bound states have a discrete spectrum of energies, e.g. cosine waves in a square well of infinite walls.
2) Scattering states, are not integrable in absolute square. They resemble very much with the Fourier functions and are even called sometimes generalized Fourier functions. The scattering waves come from $r \to \infty$ and going to $r \to \infty$ after passing through the potential well - this is why they are called scattering waves, because they come and go. Such states can be found when the potential well has a finite maximum. 
Their energies have a continuous spectrum. Many times (but not always) greater than the peak energy of the potential well.
3) Resonant states, are a class of states that comprises the metastable states of which you ask. They are in between the scattering and the bound states. A particle won't remain in such a state forever. The probability to find in inside the potential well decreases in time, in most of the cases as $e^{-\Gamma t/\hbar}$. Typical such states are the excited states of an atom. They live for a while, then they decay with the emission of a photon. When the half-life of such a state is very long (which implies small $\Gamma$) the particle, the electron in the case of the excited atom, has a very big probability to remain for a long time in this state. This is the metastable state.
The resonances also have a discrete spectrum. As to your question, they are confined, but not completely. For example in an atom, short time after an electron rised to an excited state, its wave-function is quite confined to the atom. But immediately, the wave-function begins to expand in space. Outside the atom we get a wave that all the time expands. So, after a time much bigger than the half-life the biggest intensity of the wave is out of the atom, not inside.
