Good words from Andrew Thompson:
Speaking about Methodism as a connectional movement goes all the way back to John Wesley. He referred to the group of preachers who were "in connexion" with himself, meaning that they were engaged in a common mission under his recognized leadership.
That sense of unity remained at the heart of the connectional idea after Wesley's death, when leadership moved from Wesley the individual to the conference itself. American Methodism developed a multi-layered system of conferences. Methodist preachers became identified with the annual conferences where their membership was held, but from the year 1792 all the various annual conferences began to gather in a General Conference that met (and continues to meet) once every four years. The General Conference is the fullest expression of our connection, and it alone has the authority to speak for the whole Church.
... Connectionalism means that we do not see ourselves operating as independent congregations (or pastors). We are engaged in common work. The local church is the most significant arena for making disciples (cf. Book of Discipline, Par. 120), but the local church is fundamentally connected to the whole Church's mission and ministry. Common cause is essential-- in doctrine, discipline and spirit.
Connectionalism is therefore the foundation of the entire Methodist conception of the Church. Our Book of Discipline refers to the connection as a "vital web of interactive relationships" (Par. 132) and describes each local congregation as "a connectional society of persons who have been baptized, have professed their faith in Christ, and have assumed the vows of membership in the United Methodist Church" (Par. 203). It is only because each of those congregations exists as part of a broader connection that the entire United Methodist Church can make a common witness to the world.
Methodists employ the idea of covenant when they speak of their connection. We are called into the covenant relationship of the connection. That’s the case for all members of the United Methodist Church-- both laity and clergy. Our Discipline explains this feature of the Church: "United Methodists throughout the world are bound together in a connectional covenant in which we support and hold each other accountable for faithful discipleship and mission" (Par. 125).
We live in a time when the connection is under a great deal of strain, by forces both inside and outside the Church. We can endure this period of trial and perhaps come out of it stronger than we were before. But it will require all of us to recommit ourselves to our connectional covenant. If we find ourselves unable to abide within that covenant, on the other hand, then the very integrity of the connection itself will fly apart at the seams.
The entire post can be read here.