What is a Heresy?

"Error, indeed, is never set forth in its naked deformity, lest, being thus exposed, it should at once be detected. But it is craftily decked out in on attractive dress, so as, by its outward form, to make it appear to the inexperienced (ridiculous as the expression may seem) more true than truth itself." Irenaeus Against Heresies 1.2

The word "heresy" comes from the Greek hairesis which means "choosing," or "faction." At first, the term heresy did not carry the negative meaning it does now. But, as the early church grew in its scope and influence throughout the Mediterranean area, various teachers proposed controversial ideas about Christ, God, salvation, and other biblical themes. It became necessary for the church to determine what was and was not true according to the Bible. For example, Arius of Alexandar (320 AD) taught that Jesus was a creation. Was this true? Was this important? Other errors arose. The Docetists taught that Jesus wasn't human. The Modalists denied the Trinity. The Gnostics denied the incarnation of Christ. Out of necessity, the church was forced to deal with these heresies by proclaiming orthodoxy. And in so doing, condemnation upon these heresies and the heretics became a reality.
Unfortunately, some of those who attempted to defend and establish the truth did so by killing those who disagreed with them. What would prompt such hostile actions against those who merely had "differences of opinion" on biblical subjects? The answer may not ever be fully known, but I offer this explanation.
Culturally, when Christianity arose, it arose in the midst of a hostile environment. Judaism and the Roman Empire both warred against its people and its teaching. Persecutions arose and Christians were killed for their faith. In the Diaspora (dispersion) of the late first century, Christians were scattered throughout the Mediterranean area due to the persecutions in Israel. The Roman Empire with its theology of many gods was not friendly to Christianity's monotheism. Therefore, Christians were further persecuted.
Theologically, the Bible teaches condemnation upon false doctrines and false teachers. Gal. 1:8-9 says, "But even though we, or an angel from heaven, should preach to you a gospel contrary to that which we have preached to you, let him be accursed. 9 As we have said before, so I say again now, if any man is preaching to you a gospel contrary to that which you received, let him be accursed" (NASB). See also 11 Cor. 16:22; 2 Cor. 11:1315; 1 Tim. 1:18-20; Titus 3:10. Why is this taught in the Bible? The reason is simple. Christians are saved by faith in the work of Jesus on the cross. But faith in itself is not enough. Faith is not a substance you can put in a jar. It is belief in something. Faith is only as good as who it is placed in. If you put your faith in a false God, you are lost because a false god cannot save anyone. This is why God says in Exodus 20:3, "You shall have no other gods before Me." Faith is not what saves, but faith in the true God is what saves.
I suspect that it is a combination of the cultural and theological contexts that resulted in Christians seeking to "do away with" the heretics. Heresy has the ability to damn because they have the ability to confuse the gospel sufficiently to make it powerless. For this reason, I suspect that to many ancient Christians, heresy became one of the most serious of offenses.

Essential verses nonessential

It becomes necessary to define those doctrines which separate Christian from non-Christian. It would make no sense to persecute anyone over a doctrine that is not essential to the faith. Such nonessentials, in my opinion, would include baptism of infants, pre or post-trib rapture, worship on Saturday or Sunday, musical instruments in the church, the charismatic gifts, worship styles, dress codes, etc. These kinds of subjects do not affect one's salvation. Unfortunately, the disagreements that arise around these subjects result in denominational fragmentation.
Essentials of the faith would include who God is, who Jesus is, salvation by grace, and Jesus' resurrection. From these subjects we have derived doctrines known as the Trinity and the hypostatic union (Jesus' two natures: God and man). The Bible tells us that these doctrines concerning God, Christ, salvation, and resurrection are essential to the faith. Therefore, it is of the utmost importance that Christians know their faith and know how to defend it against the doctrines that compromise the essentials.