One avenue to discover the responsibilities of eldership is to examine the Church Order. The church order has a number of articles that outline some of aspects of eldership. I have put the basic Articles of the church order in together for your information.
All officebearers, on occasions stipulated by council, classical, and synodical regulations, shall signify their agreement with the doctrine of the church by signing the Form of Subscription.
b. The elders, with the minister(s), shall oversee the doctrine and life of the members of the congregation and fellow officebearers, shall exercise admonition and discipline along with pastoral care in the congregation, shall participate in and promote evangelism, and shall defend the faith.
a. In every church there shall be a council composed of the minister(s), the elders, and the deacons. Those tasks which belong to the common administration of the church, such as the calling of a pastor, the approval of nominations for church office, mutual censure, meeting with church visitors, and other matters of common concern, are the responsibility of the council.
b. In every church there shall be a consistory composed of the elders and the minister(s) of the Word. Those tasks which belong distinctively to the office of elder are the responsibility of the consistory.
a. The council of each church shall delegate a minister and an elder to the classis. If a church is without a minister, or the minister is prevented from attending, two elders shall be delegated. Officebearers who are not delegated may also attend classis and may be given an advisory voice.
a. These are collective responsibilities. It is important to use the different gifts of members to ensure that together all the tasks are accomplished. As an individual, you can not do it all.
b. Here is a list of tasks:
i. Oversee life and doctrine
ii. Exercise admonition and discipline
iii. Exercise pastoral care
iv. Participate and promote evangelism
v. Defend the faith
vi. Common administration of the church (calling, nomination, mutual censure, etc)
c. Basically, these tasks cover the work and life of the church. However, it is not the intention of the church order that the elders do all the work of the church. Elders provide guidance and accountability in the church community as a community of elders.