What glass transition entails becomes clear when you compare the cooling down of a glassy, amorphous
material with that of a crystalline
When a crystalline material is cooled down from a temperature above its melting point to a temperature below the melting point, its structure will slowly change with the temperature. When the temperature has come below the melting point, the melt is restructured into the ordered structure of the crystal lattice, a process which is called crystallization.
Sometimes, under special circumstances, it's possible to cool a material to below its melting point without crystallization occurring. A supercooled liquid is then obtained. There is still a change of structure with the changing temperature, but this change is not as abrupt as in the case of crystallization. Upon further cooling the viscosity of the liquid becomes ever larger until at a certain point the viscosity has become large enough that the atoms in the melt are no longer capable of reordering into the crystalline structure in the time left during cooling. In the end the viscosity becomes so large that the 'structure' of the liquid is fixed and no longer dependent on the temperature. The supercooled liquid has then become a glass. This process of supercooling and finally freezing is the glass transition.
The funny thing is that almost any liquid can become a glassy material. The trick is to do the cooling fast enough. For example, by pouring a thin stream of molten aluminium onto a rotating copper wheel (that has a huge capacity for absorbing heat and moving it away from the source), it is possible to create thin strips of amorphous aluminium 'glass'. This can then be shredded and processed by extrusion, and has very special properties.
The reason there seems to be a specific range of materials that we call glass, is that these materials consist of large molecular units that diffuse very slowly anyway, and that these materials have a very high viscosity when molten. This means that the critical cooling rate needed for these materials to show a glass transformation can be reached by just removing the heat source and letting them cool bythemselves. However, if you heat these materials to a point just below the melting point and keep the temperature that high for an extended period, the basic units of the material will diffuse and slowly a crystalline structure will form. This process is known as devitrification and it's used to manufacture glass ceramics for ceramic cookers, for example.