The unit is the individual component of the larger organization that a person served with. 

In the Army, from an infantry point of view, this would be the battalion (a regiment would be divided into several different battalions – this is why it is so important to know which battalion an individual served with).  The battalion would serve respectively under a brigade, division, corps and army (these are also all units, with their own war diaries, but further removed from the ‘personal’ experience on the ground).  The battalion would also be subdivided into company, platoon and section levels; these are too small to have their own unit war diaries.  

Other formations had different basic units, and these were not necessarily the same during both world wars.  For example, a cavalry regiment would have a squadron, an artillery formation would probably have a brigade and/or a battery (although in the Second World War the brigade would be replaced by a regiment), while the Royal Army Service Corps would have companies.

In the Royal Navy, the unit would usually refer to a ship or Shore Establishment (a naval base on land).  The Royal Marine Light Infantry and Royal Naval Division were organised into battalions.