The term Calvinism comes from a pastor and teacher in the 1600's name John Calvin. He lived at the time of the Protestant Reformation, was in the second wave of reformers (Martin Luther being in the first), and is considered to be one of the greatest theologians that ever lived. Calvinists are called such, not because we worship him or follow him or anything like that, we are called that because we agree with what he taught about God from the Bible. In fact, most people who are considered Calvinists don't like to be called that, just because it can lead to all sorts of confusion. They prefer the term “Reformed” - meaning that they follow in the tradition of the Protestant Reformers.
When it comes to what Calvinists believe there are a couple of things you have to know to begin with. The first is that they take the authority of the Bible very seriously. They believe that the Bible is the infallible (incapable of making mistakes or being wrong), inerrant (without error)Word of God. The second is that they believe very strongly in the sovereignty of God, that is His right and ability as the creator of the world to rule over it in every aspect. Those two beliefs are at the core of Calvinism.
Now, on to the things that Calvinists are most known for believing. Their beliefs can be summed up with the acronym T. U. L. I. P.
The “T” in T. U. L. I. P. stands for Total Depravity. Depravity isn't a word that most people use on a regular basis, so let me define it for you – it means “moral corruption.” They believe that after Adam and Eve's sin the Garden of Eden (Genesis 3), sin effected them to their very core so that even when they wanted to do good, they cannot do it in such a way as to please God (Rom. 7). That is why most Calvinists prefer the term Radical Depravity at this point. Radical meaning that depravity has effected the radix(Latin for root) of peoples beings. This change was then passed on to all Adam and Eve's descendants so that all people naturally, that is apart from Christ, are under God's wrath (Ephesians 2:1-3).
The “U” in T. U. L. I. P. stands for Unconditional Election. Election means God choosing who will be saved. All Christians believe this in one form or another. The common form of this is something along the lines of we choose God in faith then he agrees with that and in turn chooses us. This is known as conditional election – God's choosing us is in turn based on us first choosing Him. Calvinist believe in UNconditional – that is God does not choose us based on any prior conditions, but solely because of His grace and mercy. They believe that God chose who would be saved before the foundations of the world (Eph. 1:3-14) and not based on any future action, for good or bad (Romans 9).
The “L” in T. U. L. I. P. stands for Limited Atonement . Calvinists at this prefer the term Definite Atonement , meaning that Christ died, not to make salvation possible for all people, but certain for God's elect. Calvinists derive this believe from passages such as Matthew 1:21 where it says “She will bear a son, and you shall call his name Jesus, for he will save his people from their sins (ESV).” Also, since Jesus is the Good Shepherd, He lays down his life for His sheep (John 10:11, 15). He then goes on to define who His sheep are in verses 26-27 of that same chapter: “ ... but you do not believe because you are not part of my flock. My sheep hear my voice, and I know them, and they follow me.” In other words those that come to faith in Christ do so because they are amongst Jesus' sheep, while those that don't, don't because they aren't one of Jesus' sheep.
The “I” in T. U. L. I. P. stands for Irresistible Grace . Irresistable Grace is the belief that for those whom God the Father has chosen for salvation and for whom Jesus has died, the Holy Spirit will actually bring to saving faith, in God's own timing. The derive this belief from such passages as found in the sixth chapter of John. “All that the Father gives me will come to me, and whoever comes to me I will never cast out … [a]nd this is the will of him who sent me, that I should lose nothing of all that he has given me, but raise it up on the last day (John 6:37, 39 ESV). “No one can come to me unless the Father who sent me draws him. And I will raise him up on the last day. It is written in the Prophets, And they will all be taught by God. Everyone who has heard and learned from the Father comes to me (John 6:44-45 ESV).” “... no one can come to me unless it is granted him by the Father (John 6:65).”
The “P” in T. U. L. I. P. stands for Perseverance of the Saints . Perseverance of the Saints means that those that God has chosen for salvation, for whom Christ died, and who the Holy Spirit has called to salvation, will ultimately persevere in the faith to the end, despite periods of back-sliding or faithlessness. A classic proof-text for this belief is Romans 8:28-30, which contains what some have called the “Golden Chain of Salvation.” “And we know that for those who love God all things work together for good, for those who are called according to his purpose. For those whom he foreknew he also predestined to be conformed to the image of his Son, in order that he might be the firstborn among many brothers. And those whom he predestined he also called, and those whom he called he also justified, and those whom he justified he also glorified.” In this passages we see a definite chain of events, especially in in verse 30. God predestines certain people → God calls those people → God justifies (declares them to be innocent based on Christ's atoning sacrifice) those people → God glorifies (the final stage of salvation that takes place in heaven) those people.