Quick start :
Define result values by the first method (described below), and press button "Ok".

Only "Cycle samples out" and "Cycle samples in" will be used for resampling; all other values are only for help to calculate these two values.
Resampling always adds some distortion to a sound, and to make the distortion as small as possible, here is used the two integer values (which makes vulgar fraction) way; more these values are smaller - less distortion will be added to a sound.

And here are few methods to calculate the resulting values.

By first method : input "Rate out" and "Rate in", and press "Simplify", what will make those numbers smaller if possible.
If checkbox "Auto get samplerate" is on, then "Rate in" will be automatically filled at the dialog start, getting value from a waves which are selected for resample, so you will have only to input the new sample rate into "Rate out" window.
More detailed : will be calculated "greatest common divisor" for the two input values, and then they will be divided by that divisor.

By second method : input float point values into "Relation", input necessary precision into "Tolerance", and press "Approximate".
If the result's error from the resulting vulgar fraction is too big for you, then press "More exact" button until you will get a necessary precision.

By third method : just input vulgar fraction into the output values.
For example :
(not "441" and "480", but) "160" and "147", if you would like to convert sound from DVD standard (48000) into CD standard (44100).
"2" and "3", to convert for example 48000 rate into 32000.

*the processing function actually have two subroutines, Resample Up and Resample Down, which works by absolutely different methods.

Information about resample is on Resample.