I understand what an Azeotrope is and I understand that some zeotropes can be turned into an azeotrope to properly distill or separate the compounds ( or so Wikipedia says ) but i still cannot figure out what Azeotropic drying is?
My understanding of azeotropic drying, as I have used it, is simply boiling a material in a solvent which forms an azeotrope with water and collecting the water as it condenses and separates from the azeotrope.
For example, if you wanted to dry an air sensitive compound which is an oil at room temperature then you can set up a Dean-Stark apparatus under an inert atmosphere with a solvent such as toluene to drive off the water(as described by AlphaScent). This is sometimes much faster than using a high vaccum line (or more appropriate if your substance is relatively volatile) and gives a sample which is sufficiently dry for many applications.
You can also use a solvent such as dichloromethane for heat labile substances where a Clavenger Apparatus is employed rather than a Dean-Stark.
In both of the above cases it is possible that the returning "wet" solvent causes an issue, in this case it is possible to dry the returning solvent by way of a soxhlet extractor filled with molecular seives rather than a Dean-Stark. This also works well for removing low MW alcohols such as methanol or ethanol.