Christians in the more evangelical tradition usually follow Calvinism, named after John Calvin. It was a turning point in Christian thinking, much as Martin Luther with his theses posted on the church door that led to a widescale reformation of the church. But for me personally, I can’t quite wrap my head around some of the implications.
(The other key figure is a theologian named Jacob Arminius who argued an almost opposite viewpoint to 5 point Calvinism)

The 5 main tenets of Calvinism are pretty well known

[T]otal Depravity
[U]nconditional election
[L]imited Atonement
[I]rresistable Grace
[P]erserverance of the Saint

Total depravity infers that we as a humans are completely depraved and due to our depravity we are unable to make connection with God.
Contrasted with Arminianism, we know that we are sinners, but this does not preclude us from reaching out to God and admitting that we are not in right relationship with him. In Calvinist way of thinking, only God can reach out to us because of our hardened hearts. To me this sounds more like a Pharaoh type story where God used his power in a deliberate sense. As humans I think we realise our shortcomings and have the real ability to call on God and not in a “half-duplex” way under Calvinsim.

Unconditional election. God chooses us. We cannot choose him. It is not conditional on anything on our behalf. Not our acceptance, faith, belief, understanding, repentance or anything else.

Arminianism turns this argument on it’s head. It suggests that a person can choose God. Anyone that hears the word of God, learns about Jesus and his sacrifice can make a deliberate and conscious choice to either follow him or not. In Calvinism, no-one can make a conscious decision for God, as this decision was made even prior to our existence!
In the world, we often treat winning souls using an Arminian model anyway. We try to win people to Christ, and this is done through outreach programmes, overseas missionaries, inviting friends to church etc. If God has already decided who his people are, then are all these activities really just illusory as the result is already foreknown and predestined?
Those that aren’t chosen face a life of eternal damnation for something they cannot ever rectify or make amends to. They simply weren’t chosen for whatever criteria God selects. Would God create beings for the express purpose of sending them to hell? I don’t think so…

Calvinism drilled down further has a number of theories about how we are saved. Supralapsarianism says that God chose his elect before the creation of man. Infalapsariansim says that God chose his elect as a result of the fall. To me, Infalapsariansim would sound more reasonable. It would show that God chooses to bestow some sense of forgiveness to us, even if undeserving, rather than just a forethought out plan (as Calvinism would argue) where humans were inherently designed to disobey (ie. fall) by grand design.

Limited Atonement. In Calvinism, God’s love and grace are only available to the elect. The most famous bible verse John 3:16 states that God so loved the world that he gave his only son. It did not say For God so loved the elect. Some argue that God loves the entire world but eternal salvation is only available to God’s chosen.

Irresistible Grace. God’s grace is irresistible. Nobody can resist his grace. The problem is that real life often throws a number of curveballs. Perhaps where someone who has fallen away and explicitly rejected God, only later to repent and re-establish themselves to God. Under the strict code of Calvinism, and if we are to follow these ideas in their original non-polluted sense, then if you resist his grace, the only conclusion that follows is that you actually weren’t ever a Christian.
Arminianism posits the opposite. You can fall away and be under earnest repentance be redeemed under the grace of God. It’s not a one-shot-and-you’re-out scenario.

Perserverance of the Saints. This relates to the idea that once a child of God is saved he or she is always saved. It raises a number of questions. Is a fallen believer who comes back to God covered? Under irresistible grace, no one who is truly called can become an apostate. So it seems as though this point is really just another support column of “irresistible grace”.

My conclusion: Calvinsim and Arminianism should probably only be seen as conceptual models for understanding Christianity. Though if I were to choose, I would more than likely choose Arminianism, not because it’s a “me-focussed” salvation as some might claim. It does though give the follower the conscious act to follow. It puts humans in the equation in our choice to follow God and does not reduce us to the “lottery” of election.