a moving window is usually used to filter out 'noisy' data which can arise from a variety of circumstances. The most common useage is in the area of remote sensing where the sensor response may reflect 'error' or just the inherent spatial variation that exists in the landscape. filtering with such things as the average (for interval ratio data like elevation, temperature etc) provides a dampening to spikes and provides a more generalized representation of the spatial pattern. Other filters like the most frequent, mode or dominant would be more useful for other data measured on a nominal or ordinal scale.
It just isn't the 'windowing' that is important, it is also the size and shape of the window, they all have there useages.
The other type of filter is the 'jumping' or 'block' window, that has similar functionality to the 'moving' window but it moves in predefined chunks, each chunk is the size of the specified window. There is also a link within the help section of my link that describes the block statistics as well.
Hope that gets you started about thinking about the whole idea of why one might use the focal toolset. When you want to look at stuff over time then the local toolset is the place to go and if you are interested in how 'zones' or blocks of raster data behave or their characteristics, then there is the zonal toolset which brings you into other areas such as geometry and spatial configuration.
Have fun
