Calvinism has meant different things in different contexts.  In the
broadest sense, it refers to one of the three major strands of the
Reformation (Lutheran, Reformed = Calvinist, and "Left Wing").  In
this sense it involves specific approaches to the relationship between
OT and NT, the sacraments, church government, free will and grace, and
a number of other issues.  t's a bit hard to summarize all of that in
one posting.  Among the major positions:

  the OT and NT are not completely different covenants.  Even
	those in the OT were saved by grace.  Calvin tends to
	deemphasize Law, rather than separating Law into
	ceremonial and moral (which tends to be the Lutheran
	approach).

  believes in the real presence, but contact with Christ's
	body is through the agency of the Holy Spirit.  Contrasts
	with Lutherans, who saw Christ as physically present,
	though not replacing the bread and wine, and Zwinglians,
	for which communion is just a memorial.

  church government is based on elected deacons and elders,
	with higher-level groups to provide support and
	discipline.

Calvinism historically has been associated with groups that wanted to
transform society.  While church government is often seen as a detail,
for many Calvinist groups it formed a way of creating a broad base of
trained and disciplined lay leadership, committed to implementing
Christian ideals.  It transformed Geneva, England (for a time),
Scotland, and much of the U.S.

However people often use the term Calvinism to refer to a specific set
of ideas involving justification and election.  Calvin certainly held
these views, but it's somewhat misleading to use the term Calvinism
for them, because they form only a small part of his overall system.
Here's a summary of the points involved:

1) As a result of the fall, humans are hopelessly mired in sin.  This
is often referred to as "total depravity".  This term may be a bit
misleading.  The claim isn't that everyone is an axe murderer.
Rather, it is that all parts of man are corrupted.  This contrasts
with some people who think that evil comes from our bodies, and our
minds are pure.  Calvin's point is that the mind and will are also
affected by sin.  The infection is serious enough that God can't just
offer us salvation.  If he did, we'd turn it down.

2) God in his grace has decided to save some of us.  This decision is
called "election," and those who he decides to save are called the
"elect."  In order to do this, Christ died in our place.  By God's
grace, this results in God declaring us innocent.  God's grace calls
forth faith.  Through faith, we receive justification (the fact that
God has declared us innocent) and regeneration.  Regeneration works
through repentance to bring us into conformity with God's will for us.
This is not, of course, an immediate process.  The whole process of
growing to become more Christ-like is referred to as sanctification.
Justification, in contrast, is immediate.  We are declared innocent
because Christ died for us.  He died 2000 years ago, and nothing more
is needed for justification.  Note that all of this is God's gift,
through his grace.  Even faith is inspired by God.  He does not wait
for us to have faith before justifying us.  (If he did, we would be
hopeless, since we are unable to have any real faith without his
help.)

3) God's grace is called "irresistible".  That is, we are in such bad
shape that he can't even depend upon our accepting his help.
Obedience is contrary to our fallen nature -- we want to do what we
want to do, not what God wants us to do.  So he simply gives us grace,
and starts the job of reconstructing us.  We have no choice in the
matter.  In fact his analysis is based on Augustine's, which says that
God's grace created the possibility of choice.  That is, Augustine
claims that sin destroys our ability to choose the good.  That's
ultimately what total depravity means.  We are so mired in sin that we
can't choose anything other than more sin.  That doesn't mean we can't
do good acts.  But we always do them from mixed motives: the desire to
look good, to get some reward, etc.  What we can't do without God's
help is make any decision that would truly get ourselves out of sin.
God's grace turns us around, and makes us capable of having faith in
God, and thus accepting his help.  His grace has to be irresistible,
because until it has already created faith, we would do our best to
resist it.

4) Note that as a result of this, the decision as to who is to be
saved is entirely up to God.  We are unable to have faith until he by
his grace moves us to have it.  Thus those who he calls accept
justification through faith, and are saved.  Those who he does not are
not.  This is referred to as "double predestination."  It's double to
contrast it with Luther's idea, which is called "single
predestination."  Luther says that God decides who he is going to
save.  Thus God is responsible for who is saved.  While all the others
are damned, they are damned through their own choice.  God doesn't
exactly intend them to be damned, though in some sense God is
responsible for it because he didn't give them sufficient grace to be
saved.  Anyway, that's single predestination, because properly
speaking, God only decides the fate of those who he saves.  Calvin
considers this illogical.  Clearly if God decides who he is going to
save, and the rest are damned, God is responsible for both.  Thus
Calvin says that God intends to save those who are saved, and he
intends to damn those who are damned.

5) The damned are still responsible for their damnation.  They have
disobeyed God.  It's true that he hasn't given them the grace that he
gave the elect, so he determines who is going to be left in their
sins.  But they do choose to disobey.  Thus it is just to condemn
them.

6) God's choice does not depend upon any merit of the person.  He
chooses people from all nations, all walks of life, etc.  Election is
a completely unmerited gift.

7) People have a tendency to "back-slide".  When God decides to save
someone, he grants "perseverance".  That is, while a person who he has
chosen may fail from time to time, God's grace will preserve (or
restore, if the back sliding is bad enough) his faith, and make sure
he ends up saved.  This is often called "once saved, always saved."
Of course we observe that there are people who are Christians for a
while and then fall away permanently.  There are two possibilities.
Some of them probably weren't really saved in the first place.  They
didn't actually put their trust in Christ for salvation, but simply
went along with what looked like all the Christian rules.  But others
really did have faith.  Even though they fall away, God does not
abandon those who he has chosen (though he may be mighty upset with
them at the last judgement).  There tend to be two differing
approaches to "once saved, always saved."  Some people are more
inclined to think that people who fall away probably weren't really
saved in the first place.  Others tend to emphasize that God doesn't
go back on his word, and some people will be saved even though they
really shouldn't be.  But of course, none of us deserve it anyway...

Note that Calvin does believe that people make real decisions.  Those
who are not elect actually decide to reject God.  This decision
reflects their character, which is that of sinners, and they are
responsible for it.  However in a different sense God is also
responsible, since he knows that without grace the person will choose
in this way, and he decided for one reason of another not to give that
grace.

Of course the primary point of the doctrine isn't damnation, but
grace.  What he is primarily trying to defend is the idea that those
who are saved are saved only because of the intervention of God.  He
considers it religiously important to realize that God's grace comes
before anything we do to merit it, allow it, or cooperate with it.
Nor does it come by virtue of God foreseeing that we will later merit
it, allow it, or cooperate with it.

As far as I can see, God's responsibility for the damnation of the
non-elect has less of a positive significance.  Mostly it just sort of
"falls out" as the obvious logical consequence of the other side.
Various attempts to teach a "single predestination" were being tried
at the time.  This is the idea that our salvation comes entirely from
God, but the damned are damned entirely by their own choice.  He had
seen enough of these attempts to realize that ultimately there's no
way to absolve God of responsibility for the fate of those on whom he
does not bestow grace.  Attempts to do so typically ended up creating
confusion.  Since Paul does not blush to talk of God hardening men,
Calvin sees no reason why he should either.

However the idea that God is in control of everything is important to
Calvin for another reason.  Calvin believes it's important for people
to believe that God is responsible for everything.  Ideally a
Christian should see all of life as a gift from God, even the things
that from our perspective look bad.  This only works if God's plan
includes the actions of bad people as well as good people.

Many more recent Calvinist interpretations have tended to be
determinist.  But Calvin himself does not think in a mechanistic way.
We should avoid reading Newton back into Calvin.  Jonathan Edwards, on
the other hand, does use something akin to determinism in his elegant
defense of predestination.  His book on freedom of the will is a gem
of precise definition, careful thinking, and lucid writing.  It is
certainly a consistent development of Calvin's ideas.  But I don't
think one should imagine that Calvin himself was thinking along those
lines.  Calvin never loses sight of the fact that the God works
through the personal presence of the Holy Spirit.

It is probably a mistake for Calvinists not to say a bit more about
the asymmetry between grace and reprobation than they do.  God is
responsibible for salvation and damnation in somewhat different ways.
Although God is in an ultimate sense responsible for the damnation of
those who he does not choose, Calvinists want to be very clear that
God is not the author of sin.  He is responsible for damnation in the
sense that withholding his grace makes it inevitable.  But he is
responsible for salvation in a more direct and stronger sense.  Calvin
wants to say with Paul that all the good we have we got from God.
Thus God is the actual source of our salvation and any holiness that
we may have.  He is not the source of sin in the same way.  By failing
to bestow grace, he makes it inevitable that sin will dominate some
people, but he isn't the actual source of that sin.

One of the problems with the mechanistic reinterpretation of Calvin is
that if you aren't careful grace becomes too similar to damnation.
Edwards carefully analyzes what necessity means in the context of
human lives.  His analysis is very useful as a way of explaining how a
person is responsible for his sin, but it is still an inevitable part
of God's plan.  However we have to make sure that the sort of
mechanistic thinking he uses in this analysis does not seep into our
idea of grace.  Grace is not simply a result of our heredity and
environment.  It is a personal indwelling of the Holy Spirit.  Edwards
obviously know this, but it's easier to lose track of this fact in his
work than in Calvin's.

Note by the way that I accept all of this only in a somewhat qualified
way.  I think the basic exegesis based on Rom 9 is flawed.  Paul
certainly talks about God hardening people.  But he is not thinking of
a decision by God from eternity that certain people would be damned.
Rather he is talking about temporarily delaying the Jews' acceptance
of Christ to provide an opportunity for the Gentiles to be grafted in.
I have a separate essay on predestination (file "decision") that deals
with my views in somewhat more detail.

One concern I have is that in some ways predestination ultimately
defeats itself.  Sola fide started as a way to rescue people from the
problem of salvation by works.  Luther's initial problem was that he
couldn't ever believe he had done enough: repented completely enough,
etc.  Grace was good news to him because it means that his salvation
isn't up to him.  It depends purely on God.  So his own
insufficiencies no longer are of concern.  But this only moves the
problem back one step.  How do we know that God has chosen us?
Somehow this never bothered Calvin.  I guess he was very
self-confident.  But later Calvinists had a Reformed version of
Luther's suffering.  They didn't have to worry about whether they had
done enough, but how could they be sure that they were elect?  So we
had the whole unedifying set of discussions of how to tell whether
someone is elect.  (That whole business was very much not present in
Calvin, by the way.)  As I claim in my essay on
once-saved-always-saved, there is probably no theoretical way out of
this box.  Only in the context of a relationship with God can we
become confident of our salvation.  Doctrines and tests of savedness
quickly become self-defeating.