between whom was the battle?
The Allies and the Germans.
Battle of Villers-Bocage
The Allies and the Germans regarded control of Caen as vital to the Normandy battle. 
where did the battle of villers-bocage take place? Villers-Bocage
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Battle of Villers-Bocage [SEP] Introduction
After six hours, Hinde ordered a withdrawal to a more defensible position on a knoll west of Villers-Bocage. The next day the Germans attacked the brigade box, arranged for all-round defence, in the Battle of the Island. The British inflicted a costly repulse on the Germans and then retired from the salient. The Battle for Caen continued east of Villers-Bocage, the ruins of which were captured on 4August, after two raids by strategic bombers of the Royal Air Force.