Scattering vs bound states

These terms apply when you're solving the Schrodinger equation with a potential that goes to zero at large distances. In this situation, the solutions with $E<0$ have the property that $\psi$ dies away to zero for large distance. So the particle is, with high probability, guaranteed to be in a confined region (not at large distance). So those are bound states.

The solutions with $E>0$, on the other hand, do not die away to zero at large distances -- instead, they go like $e^{ikr}$ where $k=\sqrt{2mE}/\hbar$. So these solutions represent particles that have high probability to be arbitrarily far away. Physically, they are useful when describing particles that start far away, approach the scattering center, and end up far away again. Hence the name "scattering states."

Let me explain with a simple example. Consider a particle in a finite potential well. There will be two cases:

i) $E<V$;

ii) $ E>V $.

If the energy of the particle its smaller than the magnitude of the potential, the particle will be confined in the box forever, that is, the particle its bounded to the thing that is generating the potential. In this case where the particle its confined to a finite space, in a bound state, the energy will be quantized, that is, only multiples of a certain quantity of energy will be allowed.

But if the energy of the particle it's greater than the intensity of the potential the still will "feel" the "hole" below it, a portion of the "particle" will be reflected, and will go back, and the other portion will cross the well.

The energy did not need to be necessarily smaller than zero to a bound state occur, as a matter of fact it only need to be smaller than the intensity of the potential.

All of the information described above can be obtained by solving the Schrödinger Equation to the potential in question, as was done in the link in the top of the answer.

A very good introductory book in this subject its "Introduction to Quantum Mechanics" by David J. Griffiths. Read it, it is very nice!