The ability to represent a function at hierarchical levels of resolution, or detail, and then being able to examine it at any level. The highest resolution is the complete function, and lower levels have lesser and lesser detail, and are coarser. One way to do this is using wavelets. This idea is getting very popular in computer graphics, to represent curves and surfaces, because it is useful to render them at various levels of detail.

Another interesting thing, is that every MRA can be generated by the use of recursive subdivision, which is a quite cool idea, where you take a polygon, or even a polyhedron, and split each edge at its midpoint, and then move the vertices around. The surprising thing is that if you do this process correctly you can get all sorts of nice (or weird) looking curves out of it :)

Kind of a neat really, perhaps even a topic for a Masters thesis.