A parish is the smallest unit of administration within the Anglican church. Most parishes have just one church, called the parish church. Some parishes have more than one church; this instance is usually found in areas with sparse or declining population, so that only the clergy need travel far. Parishes combine into dioceses.
"Priest" is a special term for the minister of a Roman Catholic, Anglican, or Orthodox church. Historically, the term meant someone who performed a sacrifice; later the term referred to those who said Mass. A person becomes a priest by being ordained by a bishop. Most bishops require special training for this, which is typically obtained in a theological college or seminary.
A "Province" is an administrative division of the church that is bigger than a diocese and smaller than the whole world. Many national churches are divided into provinces; for example, Canada is divided into four administrative provinces and Australia into five. And Australia has one diocese that is not in any Province; it is called "extra-provincial". In general no one cares about these provinces except church employees. The word "province" does not appear anywhere in the web site of the Anglican Church of Canada except in the minutes of the General Synod.
In some parts of the world, typically those that were never English colonies, the number of Anglicans is small enough that there are not individual national churches. The Province of Central America has several countries, as does the Province of Central Africa.
A transnational province is one that spans more than one country.
A rector is a priest who is the leader of a self-supporting parish. If the parish is not self supporting, its leader is usually referred to as a vicar.