Projection transformation

It has been confused me for a while. How to define or transform the projection of a shapefile in ArcGIS such that it can align with other maps with different coordinate system. Here is the steps in ArcGIS 9.X.

1. Figure out the original projection of your map even though it has not been defined before, namely it does not have any projection file (*.prj) along with other files for shapefile.
2. Two coordinate system can be defined in a shapefile, one is geographical coordinate system which is essentially for longitude and latitude; the other one is projected coordinate system which is for the planar coordinate (two-dimension)
3. To define a coordinate system to a shapefile, Go to ArcToolbox -> Data Management Tools -> Projections and transformations -> Define projection, and then a window will popup for us to select the shapefile (feature class) to define and the coordinate system
4. To change a coordinate system of a shapefile in order to align with other shapefile,
   
1. Determine the coordinate system to which we would like to transform
   
2. Go to ArcToolbox -> Data Management Tools -> Projections and transformations ->Feature -> Project
3. Specify the shapefile to transform (feature class), the name for the new shapefile with new coordinate system (projection), and the new coordinate system in the popup window
4. OK
5. The new shapefile will automatically be added into the current project
6. Note that the table of the new shapefile is exactly the same as the original shapefile. The new geometry values upon the new coordinate system can be generated by adding new field in the Table and do the Geometry calculation.