Setting the Coordinate System (Projection) for a Data Frame
By default, ArcMap will use the coordinate system of the first data layer added to a Data Frame for the coordinate system of the Data Frame itself. Often it is desirable to use a different coordinate system for the display of the data. This can be done by altering the Properties of the Data Frame. Doing so will cause ArcMap to have to re-project the data 'on-the-fly' from the original coordinate system of the data to the new coordinate system of the Data Frame.
In the example below the coordinate system of the data (and thus the Data Frame) is GCS_WGS_84 (or Geographic Coordinate System, which is Lat-Long, using the World Geodetic System of 1984). When using this coordinate system, note how the lines of latitude and longitude create square boxes. In reality, the lines of longitude should be converging towards the north pole. This is an example of a type of map distortion that is introduced by the coordinate system...
Right-click on the Data Frame and choose Properties to open the Data Frame Properties sheet:
Click on the Coordinate System tab
For an "un-projected" coordinate system (e.g., Lat-Long) open the Predefined / Geographic Coordinate Systems folder
For a projected coordinate system, open the Predefined / Projected Coordinate Systems folder
For a map of Europe, you might browse to the Europe folder within the Continental folder
Choose the coordinate system of choice and click OK
You can also click Apply if you wish to check out a coordinate system (it often takes multiple attempts to find one that works)
If you choose a coordinate system different from the data in the map, you will get a Warning message. This is actually what we want to happen in this case, so simply click Yes. If you wish you can check a Don't warn me again box.
The resulting Data Frame clearly shows the effect of the new projected coordinate system (note how the lines of longitude now appear to be converging towards the north pole)