Synergism is just a Greek way of saying “co-operation,” which in turn is just a Latin way of saying “working together.” Paul uses the corresponding Greek verb when he describes himself and his colleagues as “co-working” (2 Cor. 6:1) with God as ambassadors for Christ, through whom God urges people to be reconciled to himself (ibid., 5:20). Monergism, a much more recent term, means to work alone, having no co-worker. So monergists are those who think that in some respect God works alone.
The crucial question is: in what respect? The standard Protestant view is monergism with respect to justification: God alone renders us just or righteous in his sight, without our co-operation. But most Protestants would add that sanctification is a co-operative enterprise in which our will and work have a necessary role to play, working together with the grace of God. So most Protestants are monergists about justification but synergists about sanctification. And since justification by faith alone is all that is necessary for salvation, most Protestants are also monergists about salvation.
Of course in order to be thoroughly monergist about justification one must also be monergist about the faith by which we are justified, understanding it to be a divine gift resulting from grace alone and not from human work. Luther, in effect, insisted on this type of monergism when he excoriated the medieval nominalist notion of “congruent merit,” according to which sinners work to acquire the gift of “first grace” (meaning roughly, the gift of conversion and true faith) by praying as well as they can, trying their best to “do what is in them” (facere quod in se est) even without grace. The term “synergism” seems to have come into use for the position rejected by the Lutheran orthodox theologians when they reaffirmed Luther’s doctrine in the Book of Concord in 1580 (see especially article 2). Later, Calvinists used it to describe the Arminian position that our free will has an independent role to play in accepting the gift of grace. Synergism, for both Lutherans and Calvinists, means the teaching that grace does not simply cause us to have faith, but rather makes an offer of salvation which it is up to us to accept or reject. Both Lutherans and Calvinists reject this synergism, and thus can aptly be labeled monergists with respect to the gift of faith.
The question of whether Augustine is a monergist or a synergist is more complicated. For one thing, even at his most monergistic, Augustine does not deny that we are active in our own salvation. Augustine is a monergist with respect to the origin of faith, for instance, in that he sees it as resulting from prevenient or “operating grace” rather than “co-operating grace” (his terms). But for Augustine this does not take away the role of human free will, for what prevenient grace does is precisely to move our wills so that they freely will the good. Hence for Augustine grace never undermines or replaces free will. In that sense he is never a radical monergist, as if the human will had no active role to play. On the other hand, he is indeed a monergist in a less radical sense, because for him the gift of faith is wholly the work of God, since even our freely willing to accept God’s gift is a work of grace alone.
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