bound state
A bound state is a quantum state in which a particle is trapped near some region rather than escaping to infinity. An electron held by the pull of an atomic nucleus, or a particle sitting in a well of low potential energy, cannot wander off freely; its wavefunction is concentrated where the trap is and fades away to nothing far outside. Such a particle is, in everyday terms, stuck.
The most striking feature of bound states is that their energies come only in a discrete set of allowed values, not a smooth continuum. The wavefunction must fit neatly into the trap and die away at the edges, and only certain energies let it do so without contradiction. Energies in between are simply not permitted. This is where the word 'quantum' earns its meaning: confinement forces energy into separate steps.
Bound states stand in contrast to free or scattering states, where a particle has enough energy to roam off to infinity and its energy can take any value. Whether a state is bound or free depends on the shape and depth of the potential and on how much energy the particle has. Much of quantum mechanics is the study of which bound states a given trap allows and what their wavefunctions look like.
When a wavefunction must vanish far away, only certain quantized energies are allowed.
A bound state is a property of the whole system, not just the particle. The same electron is bound inside an atom but free once knocked out of it; nothing about the electron itself changes, only its situation in the potential.