The Vestry is the governing body of an Episcopal Church, or Charge. Vestry Members are normally the “charity trustees” of the charity comprising the Charge. They make decisions about the running of the Charge, some of which might have significant financial implications which is why it is such an important and responsible position.
Most Vestry members are elected at the Annual General Meeting of the Charge. Some members of the Vestry hold special positions. These are the Lay Representative and a reserve or Alternate Lay Representative. Their job is to be involved in the local grouping of Charges and Churches, called an Area Council and also to be involved in the running of the Diocese as members of the Diocesan Synod. These representatives are elected by the AGM.
The Vestry also appoints other officers, who become members of the Vestry ex officio. This includes a Secretary, a Treasurer and a Lay Chair of the Vestry.
The Rector is also a member of the Vestry and usually (though not always) Chairs meetings of the Vestry.
The current vestry and their responsibilities can be found here.