Monergism:The doctrine that the Holy Spirit is the only efficient agent in regeneration – that the human will possesses no inclination to holiness until regenerated, and therefore cannot cooperate in regeneration. Monergism is when God conveys that power into the fallen soul whereby the person who is to be saved is enabled to receive the offer of redemption. It refers to the first step (regeneration) which has causal priority over, and gives rise to, the spiritual ability to comply with all the other aspects of the process of being united to Christ, (i.e., the ability to apprehend the Redeemer by a living faith, to repent of sin and to love God and the Mediator supremely) It does not refer to the whole process that it gives rise to (justification, sanctification), but only the granting of the spiritual capacity to comply with the terms of the covenant of grace.
Synergism:“…the doctrine that there are two efficient agents in regeneration, namely the human will and the divine Spirit, which, in the strict sense of the term, cooperate. This theory accordingly holds that the soul has not lost in the fall all inclination toward holiness, nor all power to seek for it under the influence of ordinary motives.” This unscriptural view is the greatest threat to a true understanding of salvation in the Church today.
From R.C. Sproul -
Monergistic regeneration is exclusively a divine act. Man does not have the creative power God has. To quicken a person who is spiritually dead is something only God can do. A corpse cannot revive itself. It cannot even assist in the effort. It can only respond after receiving new life. Not only can it respond then, it most certainly will respond. In regeneration the soul of man is utterly passive until it has been made alive. It offers no help in reviving itself, though once revived it is empowered to act and respond.
From John Hendryx
Monergism simply means that it is God who gives ears to hear and eyes to see. It is God alone who gives illumination and understanding of His word that we might believe; It is God who raises us from the dead, who circumcises the heart; unplugs our ears; It is God alone who can give us a new sense that we may, at last, have the moral capacity to behold His beauty and unsurpassed excellency. The apostle John recorded Jesus saying to Nicodemus that we naturally love darkness, hate the light and WILL NOT come into the light (John 3:19, 20). And since our hardened resistance to God is thus seated in our nature and affections, only God, by His grace, can lovingly change, overcome and disarm our rebellious disposition. The natural man, apart from the quickening work of the Holy Spirit, will not come to Christ on his own since he is at enmity with God and cannot understand spiritual things (1 Cor 2:14; Rom 8:7).
Shining a light into a blind man's eyes will not enable him to see, for sight first requires healthy eyes or some restoration of his visual faculty. Likewise, reading or hearing the word of God itself cannot elicit saving faith in the reader (or hearer) unless the Spirit first "germinates" the seed of the word in the heart (1 Thess 1:5), which then infallibly gives rise to our faith and union with Christ. The problem here is not with the Word or God's Law but with man's corrupt and prideful heart. As an example of how the Spirit uses the means of the spoken word to disarm closed hearts, the Book of Acts records the work of the Holy Spirit during the preaching of the apostles and, in one instance, states that "the Lord opened her [Lydia's] heart to respond to the things spoken by Paul," (Acts 16:14). The Spirit must likewise give all His people spiritual life and understanding if their hearts are to be opened and thus respond to Christ in faith.
For more information on the biblical view of Monergism and the unscriptural view of Synergism, see the following links: