Unlike deacon, priest and bishop (see last month's magazine), these are all job titles. Becoming a priest at ordination is a bit like becoming a husband (or wife) or parent. It is, quite literally, a life-changing event. But once the change has happened, there are jobs to be done.
Clergy minister in a variety of situations. All clergy who work in a parish are technically a curate, that is someone who has the "cure" (or care) of the souls living in the parish. Nowadays, however, the word curate almost always means someone who is an assistant to the incumbent. The latter is the general term for someone in charge of a parish and it covers vicars, rectors and priests in charge. These all do the same job and the difference between them is historical, relating to the way they used to be paid. The Scottish Episcopal Church does not have vicars (again for historical reasons) and in the American Episcopal Church a vicar is a junior member of a clergy team in a parish.
By the way, clergy are not paid a wage or salary in return for working a specific number of hours but a stipend, that is a sum of money that enables them to give their time to ministry without needing to find other employment.
Clergy who are on the staff of a cathedral are known as canons and the senior member of staff is the dean. There are also honorary canons, a title given by the bishop to senior clergy who have served the diocese in some special way.
An archdeacon is an ancient office in the Church. In spite of the name, archdeacons are priests who assist the bishop in the administration of the diocese. In particular, their job is to be the bishop's eyes and ears in their part of the diocese and have a duty "to bring to the bishop's attention what calls for correction or merits praise". Archdeacons are also traditionally supposed to "see to the drains" - that is making sure that church property and buildings are kept in good order.
Some clergy minister to people in a particular situation - at work, in hospital, the armed forces or in prison for example. They are called chaplains. Clergy in the Diocese in Europe were originally called chaplains for the same reason although nowadays their job is often more like that of a parish priest.
PMP