You may not be familiar with these terms, but these terms represent two foundational "camps" of theology.  Within each camp there are differences, but ultimately it all boils down to "God does it all" or "God has given man a part to play, the ability to play his part, and will judge him for what he does with his moral ability and moral responsibility". 
Monergism:  Self-willed churchmen, in an effort to escape the clear Biblical teaching that man is responsible to repent, submit, believe, obey, and persevere in holiness, have attributed man's actions and abilities wholly to God's irresistible workings within them.  However, when you take their conjectures to the logical conclusions, you find the following absurdities. In the final analysis Monergism teaches that God chooses unconditionally to save some men and damn all others.  This He accomplishes by irresistible forces, and man is only the passive receiver of God's activity.  Therefore if men go to heaven, it is because God chose unconditionally to send them there; and if men go to hell, it if because God chose unconditionally to send them there. Man has no responsibility because man has no ability, free-will, or choice in the matter.  God's workings with men are irresistible and always effective.  If God wanted all men to repent, believe, obey, and live holy; and the world to be full of godliness and love, then that is exactly the way it would be.  God has foreordained all the good and evil that is happening, will happen, or ever has happened.  God planned every murder, rape, adultery, persecution, war, crime, etc. before the foundation of the world, and everything has been just according to plan ever since.  No, they won't tell you this on their websites, but it is what they believe.  No matter what slick words and phrases and miss-used Scripture they quote, ultimately they believe that God is responsible for the minute details of activity in creation (everything is predestined). This can only mean that man has absolutely no responsibility, no ability, no choice, and hence no blame or true guilt - all was planned before man, without man, apart from man; and man plays no active role.  If they reject this evaluation, then they must admit that man DOES something of his own free will which was not predestined by God (chosen by God, and therefore God's action - not man's.)  MONERGISM IS ERRONEOUS DOCTRINE.   
Gospel Synergism:  Though Monergism is a word not found in the Scriptures; Synergism is a Scriptural term and principle.  In the Greek it is Sun-ergos, from the same two Greek roots as our word "Synergism" [Syn – “together” & “ergo” work, labor].  The Greek word used in the Scriptures from whence we get our word synergism is  strong's #4904  sunergov  sunergos,  soon-er-gos'  from  “Sun” – close union  & “ergo” – work, labor.  It is found in the New Testament in the following places: 
Ro 16:21 ¶ Timotheus my workfellow,  
1Co 3:9 For we are labourers together with God:  
2Co 1:24 Not for that we have dominion over your faith, but are helpers of your joy:  
2Co 8:23 Whether any do enquire of Titus, he is my partner and fellowhelper  
Php 2:25 Yet I supposed it necessary to send to you Epaphroditus, my brother, and companion in labour, and fellowsoldier,  
Php 4:3 And I intreat thee also, true yokefellow, help those women which laboured with me in the gospel, with Clement also, and with other my fellowlabourers, whose names are in the book of life. 
Col 4:11 And Jesus, which is called Justus, who are of the circumcision. These only are my fellowworkers unto the kingdom of God, which have been a comfort unto me. 
1Th 3:2 And sent Timotheus, our brother, and minister of God, and our fellowlabourer in the gospel of Christ, to establish you, and to comfort you concerning your faith: 
Phm 1:1 ¶ Paul, a prisoner of Jesus Christ, and Timothy our brother, unto Philemon our dearly beloved, and fellowlabourer, 
Phm 1:24 Marcus, Aristarchus, Demas, Lucas, my fellowlabourers. 
3Jo 1:8 We therefore ought to receive such, that we might be fellowhelpers to the truth. 
The principle is clearly taught in the next two passages: 
Phil.  2:12  Wherefore, my beloved, as ye have always obeyed, not as in my presence only, but now much more in my absence, work out your own salvation with fear and trembling. 
13 For it is God which worketh [energeo]  in you both to will and to do of his good pleasure. 
Col 1:29 Whereunto I also labour, striving according to his working, which worketh in me mightily. 
Before the flood God says, “My spirit shall not always strive with man” -- God strove with those who resisted, and with Noah, who cooperated. When Jesus wept over Jerusalem He expressed His will as: "How often would I have gathered..."; and declared man's will to be in opposition with, "Ye would not".  Jesus was striving, but there was no synergism, only opposition. 
Gospel Synergism teaches that God created man with moral responsibility in that he has the power of choice.  He can disobey God or obey God.  God sits as the moral governor and judge of man.  God will render to every man according to his deeds. Since man fell and is under condemnation, God graciously provided a plan of salvation for "whosoever will".  God has made the provision, but man is responsible to partake of that provision.  God actively convicts, draws, and graciously seeks to persuade men through His Spirit, Word, and people; but man can resist, rebel, and obstinately choose his own destruction. Therefore, man is responsible for his rejection or acceptance of God's salvation program.  If men go to hell, they are to blame.  God does not command men to do things for which they have no ability, and then condemn them to hell for not obeying.  Men are responsible to do all that God has commanded them to do.  Responsibility can only mean they have ability.  Without ability there is no responsibility, guilt, or blame before a just and holy God.  The Bible clearly states concerning God's righteous judgment: Romans 2:6-16 "God...Who will render to every man according to his deeds: To them who by patient continuance in well doing seek for glory and honour and immortality, eternal life: But unto them that are contentious, and do not obey the truth, but obey unrighteousness, indignation and wrath, Tribulation and anguish, upon every soul of man that doeth evil, of the Jew first, and also of the Gentile;  But glory, honour, and peace, to every man that worketh good, to the Jew first, and also to the Gentile:  For there is no respect of persons with God. For as many as have sinned without law shall also perish without law: and as many as have sinned in the law shall be judged by the law;...In the day when God shall judge the secrets of men by Jesus Christ according to my gospel." 
I have read much of monergist's materials, and am continually amazed at the twisting of words and meanings to deceive the hearts of the sincere seeker, to confuse the issue by introducing concepts foreign to Scripture, and by exalting authorities other than Scripture.  The worship of academic training as the way to understanding God's Word to common men is also amusing.  They highly esteem what man highly esteems to draw away disciples after them; but miss what God highly esteems, which is holiness and humility of heart and life.  Understanding God's Word is a Spiritual matter, not an academic matter.  If one can read the Word of God and pray; and if they earnestly apply themselves in this, they will gain more understanding of the concepts of the Word than most college professors.  Obedience is the key to understanding Scripture, not only academics.  Study is important, but obedience is more important.  You can gain much good from learning the Greek, the historical setting, and the Jewish Phrases; but obeying the commands will open the door for Holy Ghost illumination, which will do more to save the soul and open the concepts of Scripture to the mind than any college studies.   
See Psalms 119:99-100... I have more understanding than all my teachers: for thy testimonies are my meditation. I understand more than the ancients, because I keep thy precepts. 
James 1:22 gives the simple way to avoid deception:   But be ye doers of the word, and not hearers only, deceiving your own selves.  If these "Scribes" and "Rabbis" would seek to obey the Word of God BEFORE they set themselves up as teachers, they would not be deceived or deceiving others. 
Ac 5:32 And we are his witnesses of these things; and so is also the Holy Ghost, whom God hath given to them that obey him. 
When speaking to a Calvinist or Antinomian (two systems of "unconditional eternal security"), you may hear them tell you they are not "Hyper-Calvinists", but Monergism is Monergism is Monergism.  Either God has chosen to create man with ability to reason and make choices of his own, or God has predestined every minute detail of man's actions and choices.  Either man has something that he himself did of his own ability, or God has done it all.  One man charged me with not making a proper distinction between "Hyper-Calvinism" and other Calvinists.  I asked him, "According to Calvinists what did God NOT predestinate?  According to Hyper Calvinists what did God NOT predestinate?"   He could not answer the question, because it obviously shows there is no real difference between the two. 
An example of their foolishness is as follows:  They accuse Synergists of saying:  "God, I give you the glory for everything, except my faith - for that is what I did, and I get the glory for it."  Now, consider their shallow thinking:  Say I have a wife that is so pure, honest, faithful, and wise that I have the utmost confidence in her.  If I have faith in her - is that to my glory or hers?  Can I boast of the faith I have in her as being to my glory?  If I boasted of my faith in her it would only tend to her glory, not mine.  She is the one who initiated the faith, not because she controlled me, but because she won my confidence by her conduct.  Now, If you don't have faith in her, that only reveals your ignorance - you don't yet know her very well.  If I, due to selfish lust and sin in my own heart, decide to reject the light I have and leave her and slander her good name - I obstinately reject the faith and praise due to her; and instead try to justify myself by slandering her character - Is this ultimately to her shame or mine?  Having faith in her is my duty to one who has earned it.  Having faith in her only reveals that I am rightly considering the evidence before me.  This is the same with faith in God and His Word.   When I have done that which is my duty to do, I am instructed to say, "I am an unprofitable servant, I have only done that which is my duty to do".   
To believe God with full confidence is our duty, and anything less is trespass - because He has so fully manifested His righteousness in all that He has done and created (Romans 1:18-21).  For man to boast of his faith in God, as something which would bring glory to the man, is the utmost folly.  Therefore, for monergists to assert that, if "faith" was our own act, we would then boast of it, and get the glory for it, only reveals their lack of understanding.  Romans 1:20 says that man is "without excuse" for not believing in God.  Surely Monergism would be a good excuse!  Predestination would be a good excuse!  If I have free will, and God has given plenty of evidence, then I am truly without excuse for not exercising faith in Him.   
Much of Monergist's terminology actually surrenders their case.  While claiming Monergism, they actually teach Synergism.  Toplady in his hymn "Rock of Ages"  says, "nothing in my hand I bring, simply to the cross I cling"; but this is Synergism.  "To the cross I cling"  - now is that God's doing or man's?  Synergists don't believe they atone for their own sins either, but we do believe we must cling to the cross of Christ for His atonement.  Just because the man doesn't have anything in his hand to bring, he, to the cross, still has power to cling.    
In Philip Schaff's History of the Christian Church, while speaking of Augustine (354-430), whom many believe to be the father of Monergism, he says of Augustine's autobiograghy, " it every Christian my bewail his own wanderings, despair of himself, throw himself unconditionally into the arms of God, and lay hold upon unmerited grace."   Look at all the things that this person does:  bewails his wanderings, despairs of himself, throws himself unconditionally into the arms of God, and lays hold upon unmerited grace.  This is nothing but the teaching of Synergism.  A Monergist would have to say, if he were honest, "God made this person bewail the wanderings that God himself predestined and caused, God made him despair of the inability God chose for him, then God made him throw himself unconditionally into the arms of Jesus, and thereby God made him feel he received grace when only he received the next step of predestination."  It really isn't grace, being that God need not extend grace to himself, nor is it gracious to accept that which I myself chose and did.   
Monergism robs God of His grace, longsuffering, wrath, and love:  I say this because if monergistic predestination is true, then this world is a drama and God is every actor on stage.  It is like a chess game and God is playing against himself.  His love is to himself, He is longsuffering to himself, gracious to Himself, and angry at Himself -- This cancels out His attributes as false and hypoctrical.  To be patient with my own design and program is ridiculous.  To be angry at what I chose and did is foolish.   Hence, we see that Monergism is a myth, and Monergists don't really believe it.