The Archbishop of Canterbury is the Primate of All England, that is the head of the Southern Province of the Church of England. The Archbishop of York is the head of the Northern Province and merely the Primate of England, the Archbishop of Canterbury is therefore granted precedence and at least nominally, considered as the leader of the Church of England as well as being the leader of the Anglican Communion around the world.
The Archbishops of York and Canterbury are together, the joint Presidents of the General Synod, and the joint Chairmen of the House of Bishops and the Crown Appointments Commission, which is to say that they basically run the Church of England.
The Archbishop is also the Diocesan Bishop for Canterbury in Kent, although most of the work gets done by his assistants the Bishop of Dover and the Bishop of Maidstone.
The Archbishops of Canterbury to date
The first Archbishop of Canterbury was of course Augustine, Prior of St Andrew's Monastery in Rome, dispatched by the Pope Gregory to convert the heathen Anglo-Saxons. (Aided, no doubt, by the fact that the Frankish Bertha, wife and queen to Aethlebert, king of Kent, was already a Christian).
The current incumbent is due to retire and the process of selecting a replacement is currently in progress, complete with the usual bickering and accusations of dirty tricks.
Sourced from (where else?) http://www.archbishopofcanterbury.org/