Is separation of variables possible?
Assume solutions of the form Ψ(r,t) = Φ(r)ζ(t) exist.   
Inserting Ψ(r,t) = Φ(r)ζ(t) into the Schroedinger equation we obtain
(iħ/ζ(t)) ∂ζ(t)/∂t  = (1/Φ(r))[-(ħ²/(2m))∇²Φ(r) ] + U(r)Φ(r).
The left hand side of the equation is a function of t, independent of r, and the right hand side is a function of r, independent of t.  Both sides must therefore be equal to a constant, which we call E.  The left hand side then yields

iħ∂ζ(t))/∂t = Eζ(t).   ζ(t) = A exp(-iEt/ħ) = A exp(-iωt).

Here ω is the frequency of the wave function solution, and the de Broglie relation identifies E as the energy of the particle. 
The right hand side yields H Φ(r) = E Φ(r), with  H = -(ħ²/(2m))∇² + U(r) being a differential operator, or
∇²Φ(r) + (2m/ħ²)(E - U(r))Φ(r) = 0.

Summary
If U = U(r), then solutions of the type Φ(r)ζ(t) exist.  They are called stationary solutions or stationary states.  They correspond to states in which the particle has a well defined energy E = ħω. 
Φ(r) is a solution of HΦ(r) = EΦ(r), the time-independent Schroedinger equation.  It is an energy eigenfunction, since the time-independent Schroedinger equation is an eigenvalue equation of the linear operator H.  The allowed energies are called the eigenvalues of H.

If solutions Φ_n(r) can be found, for different values of E_n, then the linear combination 
Ψ(r,t) = ∑_n c_nΦ_n(r)exp(-iE_nt/ħ)
is also a solution of the Schroedinger equation, because the Schroedinger equation is a linear equation.
However, such a linear combination does not represent a particle with a well defined energy.