These four options define how to scale the image. Each option describes an algorithm used to do this. See image sampling.
None: The nearest-neighbor algorithm is used. There is no smoothing after scaling.
Linear: Touching pixels average their values.
Cubic: Touching pixels average their values so central pixels maintain the most value.
Lanczos: Pixels are passed into an algorithm that averages their color/alpha using sinc functions (similar to sine interpolation, somewhat like cubic).
None of these algorithms are directly superior, as the links describe. Instead, it would be better to list the situations in which you'd most like to use one over the other:
None (nearest-neighbor): Use when you want absolutely no sampling (blurring) of the image.
Linear: Use when you have very small text; cubic interpolation is usually better otherwise. This produces blurred, but jagged, edges.
Cubic: Use for most images. Unless the image is very small or incredibly detailed, cubic and bicubic interpolation helps keep edges smooth. According to Wikipedia, it can sometimes increase perceived contrast or cause artifacts.
Lanczos: This interpolation method is much like cubic except that instead of blurring, it creates a "ringing" pattern. The benefit is that it can handle detailed graphics without blurring like the cubic filters.
I prefer nearest-neighbor for pixel art, linear or lanczos for small text, and cubic for everything else. These choices are subjective and by no means the 'proper' usages for the algorithms.