"1st battalion Royal 22nd Rediment C company 7th platoon" describes an army unit, from the largest army grouping that they are in (a battalion) to the smallest (a platoon). "NATO-led coalition" describes what side in the conflict they are fighting on. A battalion has about 1000 men. A regiment has about 500 men. A company has about 150 men. A platoon has about 50 men. Other military unit terms in English include "division" (about 10,000 men), "brigade" (about 5000 men), "squad" (about 10 men), and "fireteam" (about 5 men).

It's an army grouping, ground forces (British or U.K. commonwealth military, indicated by the 'Royal'). 'Coalition' is a cooperating [allied] campaign of different countries' military forces. NATO = North Atlantic Treaty Organization (Countries mainly in Western Europe and North America) So, I think that this is talking about a British army group working as part of NATO in Afghanistan.