Let solve a PDE, if it happens that , i.e. any linear combination is a solution, then we say the PDE is linear.
For example, consider the wave equation,
Confirm that if are solutions then is a solution as well.
Another way to think of linear is that , the solution, is just like what linear means in an ordinary differencial equation. So is linear. But is not, because by analogy we never have being considered linear ODE.
Quasi-linear means (this is how I understand it) that the highest order terms are linear.
Is not linear but it is quasi-linear. Because the highest order terms are linear.