I have created a shapefile in ArcCatalog by right-clicking the folder I want it in, then selecting new, then shapefile, then changing the type to "polygon", and using the British National Grids under the projected co-ordinate system. I then tried to add the shapefile in ArcMap by clicking the Add Data button and selecting the shapefile I created. My problem is that when I open the shapefile in ArcMap, the map is still blank and I cannot see the projected shape of the British National Grids I need. Please help.
Specifying "British National Grids" in the projection sets the projection of your data (regardless of what data your shapefile will contain). It DOES NOT populate the shapefile with vectors/polygons of the national grids.
In order to see the British National grids as polygons, you need to either find a source that already has the data, or digitize the grids yourself. The answers to another GIS SE question include possible sources for the data.