A simple explanation of the interpolation problem (ignores commonly used functions such as inverse distance squared, however):

(Source: http://www4.ncsu.edu/~hmitaso/gmslab/viz/interp1d.html)

Surfaces created from the same data.

Source: Lubos Mitas and Helena Mitasova (http://www4.ncsu.edu/~hmitaso/gmslab/viz/sinter.html)

The exponent associated with inverse distance weights determines how quickly the influence of neighbouring points drops off. With inverse distance, when trying to estimate an elevation (e.g.) at an unknown location, the weight (or significance) the interpolating routine gives to the elevation value at each known (sampled) location decays rapidly as the distance between the known and unknown points increases. How rapidly this occurs can be adjusted to best fit your real world situation by adjusting the value of the exponent. This graph illustrates the influence on inverse distance weighting for exponents 1 - 6. Note that you can use an exponent of less than 1, although that would give points further away a greater influence than nearer points! Typically weights of 1 or 2 are used.

Below is a comparison of how different sources of DEM data--at different resolutions but for the same area--differ significantly in their representations (posted with permission of Dave Patton, original source: http://members.shaw.ca/davepatton/demcompare.html)

Download this presentation by Dr. Waldo Tobler--the person who developed pycnophylactic interpolation or reallocation.

A simple graphic example of pycnophylactic interpolation (note how the transitions between the areal units becomes smoother, and also how the volume within each areal unit remains constant [i.e., if some of the blocks within an area get taller, others will get shorter to compensate]).