All materials are composed of the same basic stuff--atoms and their electrons. Atoms come in 118 different types, giving rise to enormous variation in material properties. For example, aluminum conducts electricity; add some oxygen and you get insulating aluminum oxide. One is shiny; the latter is whitish and dull.
Just like changing the kind of atoms in a material alters its properties, topological differences can lead to vastly different physical phenomena. The common analogy to clarify the meaning of topology is the comparison of a bagel to a sphere. The distinction is that a bagel cannot be deformed into a sphere without removing the hole. In this classification, a bagel can be deformed into a coffee cup and is therefore topologically equivalent.
In the context of much of the condensed matter quantum research, topological refers to the energy bands. For instance, insulators and conductors have energy diagrams cannot be deformed into one another without opening or closing gaps. Just like removing the hole from the donut makes it a sphere, here the band structure must fundamentally change to go from insulator to conductor.
The effects of topology can extend beyond geometric classification. Physical states characterized by their topology survive in the presence of disruptions. Topology offers this protection because it is a kind of global aspect of the system. Adding raisins to the bagel does not remove its defining quality--the hole in the middle.
Quantum systems are fragile; various quantum effects can be disrupted if the participating particles interact with the outside world. Topology can protect quantum states from these disruptions and thus aid the advent of quantum computation.