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Why doesn't the projection defined in ArcCatalog change the projection of data in ArcMap?
Defining a projection in ArcCatalog for your data simply adds projection information to your data that can be used by ArcGIS. Defining a projection does not transform the coordinates of your data, and therefore your data will not appear differently in ArcMap.
Defining a projection for your data is however a very important step when working with data in ArcGIS. By defining a projection, you enable ArcMap to properly overlay your data with other data, perhaps in a different projection. Defining a projection for your data also allows you to project your data to any other projection on the fly in ArcMap.
For shapefile and coverage data, projection information is stored in a .prj file, a text file that is stored with the data. For Geodatabase data, this information is stored internal to the geodatabase.
You can also define a projection for your data using ArcToolbox's Define Projection Wizard.
If you want to transform the coordinate geometry of your data to a different coordinate system (reproject your data), use ArcToolbox's Project Wizard.