If I had to define Calvinism in my own words (that is avoid using the TULIP acronym), I would have to single out two basic beliefs: the central doctrine (i.e. the Calvinist view of divine sovereignty) and the defining doctrine (i.e. unconditional election). I'll start with the latter.
By defining doctrine I don't mean a belief which forms the foundation for the system, but rather the main doctrine (or doctrines) which distinguishes the system from other theological systems. For Calvinism I would say that is Unconditional Election. If you believe in it, then you are some form of Calvinist.
Simply defined, God chooses who He will save from humanity based purely on His own desire for them, and completely apart from any attribute or accomplishment that they may have. On the human end, it can be understood that the only true difference between the elect and the reprobate is that God chose the elect.
It is important to distinguish this from the idea that we are simply ignorant of God's reasons. While some people may express election in this way, classic Calvinism has always insisted that what makes election unconditional is that there exists no feature of the recipient which is a contributing factor in God's decision.
So how does this election work? Let us take John. John is elect. When John was born, he was born Totally Depraved, just like everyone else. Then at some point, within God's timing, God regenerates him. Regeneration comes as a kind of package with exact features. First it reshapes John's will and eternal make up. Second, it forms faith within him. Third, due to John's faith, he becomes justified, and Christ's redeeming blood is applied to him. Fourth, the Holy Spirit comes within him, and dwells within him. This all happens in an instant.
This is currently my understanding of how unconditional election works.
To discuss the specific view that Calvinists have of sovereignty, I would like to take the time to look at Calvinist epistemology (the study of how we know what is true and what isn't). Like all Christian theists, Calvinists define right and wrong, reality and fiction, based off of God's attributes. Goodness, faithfulness, graciousness: they are all perfectly and properly expressed by God.
Where Calvinists diverge from some other Christian positions is the manner in which they define God's attributes. Calvinists believe that to express a divine attribute correctly, it must be emptied of all its earthly referents and understood purely. We can call this an idealistic understanding of the attribute. This is generally accomplished by first simplifying the concept to its most basic understanding, and then insisting that this simple understanding must be expressed in God in the most absolute sense possible.
For an example, let us consider grace. Grace is generally understood within the context of someone doing something which offends us, but we act positively towards them. So if grace is to be understood ideally, we must first strip away this context, and understand it abstractly. So grace is then understood as undeserved merit (so far so good). Therefore, if grace is to be expressed purely, then it must follow that the recipients of that grace are as undeserving as possible. Thus you get Total Depravity, Irresistible Grace and Unconditional Election.
Now let us take Calvinism's central doctrine: God's sovereignty. By central doctrine, I mean that belief that a system is built around in order to protect it, or establish it. Sovereignty means that one rules over a particular domain. In God's case, that is all of reality. So if sovereignty is to be understood ideally, we must first strip away the context of an earthly king, and understand it abstractly. So sovereignty is then understood as control, and the unthwartability of one's will. Therefore, if sovereignty is to be expressed purely, God must control every single detail of what occurs, and nothing happens which God doesn't want.
The purpose of this is I never want to strongly debate something without taking the time to understand the other side. So for all you Calvinists out there, is there anything which I have misunderstood?