There are two tools in ArcGIS Desktop associated with using coordinate reference systems that often cause confusion. I believe these confusions are mainly sourced from the semantics surrounding projections and the tools in question.
The tools are the Define Projection and Project tools.
In a nut shell...
Define Projection: This tool overwrites the coordinate system information in a dataset
This means only use the Define Projection tool on a dataset that has no coordinate system applied or to a dataset you know for certain has the wrong projection.
Project: This tool projects a dataset from one coordinate system to another through transformation or conversion
