Calvinism and Arminianism are doctrines attempting to understand the soveriegnty of man and the sovereignty of God. (please correct me if I am wrong)
Calvinism can be understood fairly basically with the acronym, TULIP. See image below.
Assuming my understanding is correct...
Total Depravity is basically stating, that we are so completely inebriated by our sinful nature, and hating of God in our natural state (based on Original Sin doctrine) that we are not capable of seeking God in said state. God has to change the way we work/think.
Unconditional Election is basically that, God choses who to save.
Limited Atonement, Christ died only for the saved.
Irresistable Grace, when God choses us we can't resist.
Preservation of the Saints, this is also known once saved; always saved. Basically, if all the above is true, then you are chosen by God and cannot resist God and therefor you can't not be a saved person.
Arminianism....
Free Will, humans are free to accept or reject God.
Conditional Election, we are "elect" based on our Free Will.
Universal Atonement, Christ died for alll, even those that go to hell.
Resistable Grace, because of our ability to chose to accept or reject God, we can reject his "softening of the heart" as it is often called.
Preserverance of some Saints, because we can chose, some saved will make it, some won't. Basically, you can lose your salvation.
Explanation for a 15 year old.
There are bible verses to prove/disprove either position based on where you stand. That's not what I'm concerned about so I did not post any, just explaining the two.
Please correct me if I am wrong.