The three resampling methods; Nearest Neighbor, Bilinear Interpolation and Cubic Convolution, determine how the cell values of an output raster are determined after a geometric operation is done. The method used depends upon the input data and its use after the operation is performed. Nearest Neighbor is best used for categorical data like land-use classification or slope classification. The values that go into the grid stay exactly the same, a 2 comes out as a 2 and 99 comes out as 99. The value of of the output cell is determined by the nearest cell center on the input grid. Nearest Neighbor can be used on continuous data but the results can be blocky.
Bilinear Interpolation uses a weighted average of the four nearest cell centers. The closer an input cell center is to the output cell center, the higher the influence of its value is on the output cell value. This means that the output value could be different than the nearest input, but is always within the same range of values as the input. Since the values can change, Bilinear is not recommended for categorical data. Instead, it should be used for continuous data like elevation and raw slope values.
Cubic Convolution looks at the 16 nearest cell centers to the output and fits a smooth curve through the points to find the value. Not only does this change the values of the input but it could also cause the output value to be outside of the range of input values (imagine a sink or a peak occurring on a surface). This method is also not recommended for categorical data, but does an excellent job of smoothing continuous data.