It's no secret that the present generation of young Christians in America have an increased interest in the doctrines of grace, commonly referred to as Calvinism. These doctrines generally teach the absolute sovereignty of God. They reject man's absolute autonomy, in favor of an autonomy that is compatibly with absolute sovereignty. They declare monergism: the doctrine that Salvation is of the Lord alone. These doctrines also are distinguishable from Arminianism on five points, commonly identified by the acronym TULIP.
T = Total Depravity. This is not (as commonly misperceived) a doctrine that men are as bad as they possibly can be, but that man by nature is depraved throughout, is the natural enemy of God, and is spiritually dead. Total Depravity teaches that apart from grace, man is unable to repent and believe on the Lord Jesus Christ.
U = Unconditional Election. This is not (as commonly misperceived) a doctrine that God randomly chooses men for salvation. It is a doctrine that election for salvation is not conditioned on what man does, has done, or will do. Instead, election is according to the grace of God based on the love of God (sometimes described in Scripture using the term "foreknowledge"). Election is of particular men, not of a formula. It is merciful: no one deserves to be elected. For example, election is not based on foreseen faith, or foreseen ability to have faith. Election is based on the purpose of God, the purpose being to glorify some of the human race.
L = Limited Atonement. There are usually no great misconceptions regarding what this means in itself. It means that Christ died for the elect and not for the reprobate. It teaches that Christ's sacrifice was perfect, and that consequently those for whom Christ died will certainly be saved. Contrary to its critics, the doctrine does not teach that those form whom Christ died will be saved in the absence of further means of salvation, such as the preaching of the gospel and faith in Christ.
I = Irresistible Grace. This is not (as commonly misperceived) a doctrine that God's grace forces men to do something that they do not want, that is to say, that God coerces men to act contrary to their wills. Instead, this doctrine teaches that God transforms man's will, giving man a new nature, enlivening him spiritually, opening the eyes of his mind to the true light of the gospel, and consequently persuading man of his own sinfulness, his need for a savior, and the identity of the true Savior. Irresistible grace is irresistible, not because it overwhelms the will, but because it circumvents the will. It does not approach a man as we do. We can appeal to man's desires, but we cannot change a God hater into a lover of God. This doctrine teaches that God can do so, and does.
P = Perseverance of the Saints. This is not (as sometimes misperceived) a doctrine that once man says a certain prayer, or comes to the front of the church, then afterwards he may live in the utmost debauchery and still have confident hope of eternal life. This doctrine does not assert that believers persevere unto glory in the absence of means. It is, instead, a doctrine that God finishes what he starts. This doctrine explains that God, who is able to keep us from falling, actually does do that, because He loves us. We don't fall, because God holds us up.
These are the doctrines commonly, conventionally, and classically referred to as Calvinism. Any interesting phenomenon has happened, though, recently. A number of writers who are not Calvinist in the conventional sense have desired to be called Calvinist, for one reason or another. Some anti-Calvinists will find this hard to believe. After all, it is still the case in certain Arminian and/or Pelagian churches that the label "Calvinist" is essentially an epithet. Indeed, some who embrace the doctrines of grace are uncomfortable calling the doctrines of grace "Calvinism" precisely because it - for some hearers - triggers a knee jerk reaction of antipathy.
Still, some wish to be under the name and umbrella of Calvinism, without actually holding to incompatibilism, monergism, and/or one or more of the five points. These men (and they are, so far as I have seen, mostly men and boys not women or girls) generally fall into one of several camps.
1. Synergists e.g. Semi-Peligian/Arminian/Semi-Augstinian (call it what you like)
Some members of this group refer to themselves as moderate Calvinists. The classic work in this camp is Dr. Noman Geisler's "Chosen but Free," in which Geisler (who is - in essence - an Arminian - brief C.V.) calls himself a "moderate Calvinist" and calls those who advocate conventional Calvinism "extreme Calvinists." Dr. James White has written an excellent, and quite readable, rebuttal entitled, "The Potter's Freedom." Dr. Geisler does not actually seem to hold to any of the five points of Calvinism except, perhaps, something that could be considered a doctrine of perseverance of the saints (and some Arminians will be quick to point out that a denial of Perseverance of the Saints is not essential to Arminianism).
Of course, not all Arminians are calling themselves Calvinists, and it would be better for everybody if labels had meanings such that "Calvinist" means one thing and "Arminian" means something else. That's why truth in labelling is important.
2. Amyraldians (often called "four point Calvinists") (see the discussion here)(and this bit of satire)
This group calls itself "classical Calvinists." So far, I have seen quite few members of this group. One particularly outspoken advocate of this label so far is David W. Ponter (brief CV). He seems bent on promoting the idea that although he denies the doctrine of Limited Atonement he is nevertheless to be labeled a Calvinist. He seems to refer to conventional Calvinists as "ultra-Calvinists" (and sometimes seems to suggest that some may actually be "hyper-Cavlinists") and seems to have borrowed something of Dr. Geisler's views on the particular point L (Limited Atonement) of the five points. As far as I can tell, DWP (unlike Geisler) is monergistic, incompatibilist, and so forth. DWP seems to agree with Dr. Geisler's claim that Calvin himself did not teach the doctrine of Limited Atonement.
Of course, not all four-point Calvinists / Amyraldians call themselves Calvinists, and it would be better for everybody if labels had meanings such that "Calvinist" means one thing and "Amyraldian" means something else. That's why truth in labeling is important.
3. Incompatibilist Monergists, i.e. Hyper-Calvinists
Hyper-Calvinists are those who affirm monergism (And generally the five points) but deny compatibilism. Thus, hyper-Calvinists typically deny human responsibility. This is ordinarily practically manifested in a failure to appreciate that God works through means. Hyper-Calvinists, therefore, normally do not bother to evangelize, because they do not see the point. Obviously, hyper-Calvinists agree with Arminians on the issues of compatibilism and yet recognize that salvation is completely from God. It's hard to be definitive about a complete set of views for hyper-Calvinism (though this site tries), because - as you can imagine - a lack of interest in proselyzation (here's an example) usually decimates a church within a few generations.
Occasionally, a few other groups that should probably be considered Calvinist have been labeled "hyper-Calvinist," such as those who refuse to use the phrase "common grace" or who feel the phrase "free offer of the gospel" is inappropriate.
And it is actually this fuzzy area of calling "hyper" those who refuse to use the phrases "common grace" and "free offer of the gospel" that is one of the crutches used to hold up the mislabeling by the first two groups. One sometimes will see the folks in those groups suggest that conventional Calvinists are "hyper," not because they actually refuse to use the phrases "common grace" and "free offer of the gospel," but because in the person's opinion, the conventional Calvinist position does not correspond to what the phrases "common grace" and "free offer of the gospel" mean.
The other area, as noted above, is the appeal to Calvin, particularly with respect to the five points. Of course, the Arminian controversy that led to the Synod of Dordt and the creation of the "five points" was after Calvin's death. Calvin never heard of the acronym TULIP. But, for whatever reasons, the mislabelers think that it is valuable to their labeling case to appeal to certain ambiguous comments from Calvin in order to assert that Calvin held to their particular view, and not to the conventional Calvinist view.
It seems, to this author, that the attempt is a shot a legitimacy. Apparently there is a feeling that it will be persuasive to use the "Calvin says so" argument, either in substitution of - or in addition to - a Biblical apologetic. These folks may even win some points with those who have read and loved Calvin's "Institutes of the Christian Religion." In the end, though, those who spend much of their effort trying to crawl under the mantle of Calvin will be disappointed.
We do love Calvin as a brother and father in the church. Nevertheless, it is not what he says, but what Scripture says, that matters. We do not say, "You must believe in the five points because Calvin said so," but rather "because Scripture says so." It is the doctrine of Scripture, not of Calvin the man, that matters.
We could be happy to call ourselves "Biblicists" to emphasize that fact - but that label would not convey conventionally the information that the conventional label "Calvinist" conveys.
Labels are useful, and the use of unconventional labels can be deceptive. Christians, regardless of the details of their doctrine, should be careful not to misuse labels with an intent to deceive. Christians should also be careful about using conventional labels in ways that are likely to confuse. Am I accusing Dr. Geisler of trying to trick people? No. I don't think that was his intent. I think he was trying to be ecumenical: trying to bridge the divide between Calvinism and Arminianism. Still, what he did (and what others who seem to be imitating him are doing) is to muddy the waters.
I think the Arminians among us can appreciate this problem, for their own conventional label has taken on a very broad range of meanings, including everything from what Arminius actually taught, to what Wesley taught, to what Finney taught, to what Fallwell taught, and even to what Dave Hunt teaches. (Cf. this article, link provided for contrast, not because I fully agree with it)
So far, incompatibilism, monergism, and TULIP have been usefully summarized with the label "Calvinism." Let's try to keep it that way. But even more importantly, if Calvinism is what Scripture teaches, then let's believe it: if not, let's abandon it.
Sola Scriptura (Scripture Alone) and Sola Deo Gloria (To the glory of God alone) not Solus Calvinus (Calvin Alone).
UPDATE: I've updated the groupings a bit. Notice JCT's objections in the combox, based on issues relating to prevenient grace.
UPDATE: Also, this article (link) provides a different perspective on the labelling issue, and one that also should be considered.