Regiments & Corps
The British Army is not really a single organisation but rather an amalgmation of a vast number of disparate pieces. In recent years this has become less so but the loyalty of any soldier or officer is still fundamentally to their Regiment or Corps.
Those units whose job is to engage in the direct fire battle with the enemy:
Combat Support Arms
Those units who provide direct battlefield assistance to the teeth arms:
Combat Service Support Arms
Supporting organisations without which no military activity could take place: