Q:

ArcGIS feature has both a geographic and a projected coordinate system

As far as I understood a shapefile is either defined as being a Projected Coordinate System or a Geographic Coordinate System. (Seeing as it is just a bunch of numbers and a coordinate system/metadata to define what they mean)
However, in ArcMap, some of my shapefiles have both projected and geographic coordinate systems defined. (See quoted section below).
What's going on here? And how can I define the projected coordinate system for another shapefile that is only currently defined by the geographic coordinate system without completely converting it and losing the geographic coordinate system?
I find it hard to understand why geospatial processing functions don't all have in-built conversion methods. I'm pretty sure when I was using QGIS it just handles all the coordinate system stuff in the background.
The Layer Properties, Source tab:

Data Type:  Shapefile Feature Class 
  Geometry Type:  Polygon.
  Coordinates have Z values:  No .
  Coordinates have measures:  No .
Projected Coordinate System:  British_National_Grid .
  Projection: Transverse_Mercator .
  False_Easting:  400000.00000000 .
  False_Northing: -100000.00000000 .
  Central_Meridian:   -2.00000000 .
  Scale_Factor:   0.99960127 .
  Latitude_Of_Origin: 49.00000000 .
  Linear Unit:    Meter .
Geographic Coordinate System: GCS_OSGB_1936 .
  Datum:  D_OSGB_1936 .
  Prime Meridian:     Greenwich .
  Angular Unit:   Degree .

A:

Making it short and simplifying a bit:
A projected coordinate system comes (optionally) on top of a geographic coordinate system. 

A geographic coordinate system is a model of the Earth's surface (spheroid) and it uses angular units (Lat/Long coordinates).
A projected coordinate system is the projection of a spheroid model on a flat surface and it uses distance units - meters, feet, etc. - (XY coordinates).

When projecting data between 2 coordinate systems (projected or not) that are based on different geographic systems, you need to choose a transformation. For some pairs of systems, you have several transformations, some being more recent and thus more accurate, some more approriate for certain area, and so on. So the software should better not choose this for yourself, you should decide based on your data and needs.
More info about this topic in ArcGIS Desktop help: read e.g. Projection basics for GIS professionals.

