In the previous recipe, you saw the same
NUTS_RG_03M_2010 feature class that has the ETRS89-GRS80 geographic coordinate system in nine data frames that have different projected coordinate systems. The projection of the feature class was made on the fly by ArcMap based on every data frame's projected coordinate system.
To permanently transform the coordinate reference system of a feature class, use the Project tool from ArcToolbox. This will create a new feature class with the new coordinates for the features. ArcGIS offers multiple versions of a transformation between different datums. It's important to choose the optimal transformation because it will affect the coordinates' accuracy.