Libertarian free will is the concept that men and angels have the ability to make real choices that have not been pre-determined by God. Arminians believe in free will, while Calvinists generally do not.
The Arminian belief in free will is rooted in our understanding of the goodness of God. We believe in free will not because we are interested in usurping God's authority, but instead because we want to protect God's character. We also recognize that free will comes directly from the hand of God. Man has free will because he is made in the image of God. To the extent that man can make any decision on his own, it's only because God has given man that ability, because it pleases Him to do so.
Calvinists instead believe in determinism (or compatiblism) - that from the beginning God has decreed everything that happens. The problem with determinism is that if God has decreed everything that happens, that would logically make Him responsible for evil. Scripture is very clear that God is not responsible for evil. Evil is a result of the disobedience of men and angels. Logically, if there were no free will, there would be no evil. Calvinists are against the concept of free will because they want to protect the sovereignty of God. They argue that if God is truly sovereign then man can't have the final say. They believe that free will makes man independent of God, and if man is independent of God, then God is not sovereign.
It is helpful to recognize that this issue of free will is a secondary concern for both Arminians and Calvinists, and flows from our understanding of who God is. Disagreement over the issue originates from honorable intentions in both theological systems. Each side wishes to protect an aspect of God's character, and each side believes that the other side's understanding does damage to the character of God. Arminians believe that the goodness of God requires that man have free will in order to explain evil. Calvinists believe that the sovereignty of God requires that there can be no such thing as true free will, or else God is not really in control. Given that backdrop it is important to understand the motivation of Arminians, and that we do affirm the sovereignty of God. However, we define sovereignty in a slightly different way. This enables us to affirm both free will and sovereignty.
A simple definition of sovereignty is: God does what He pleases. Arminians believe that true sovereignty is based on two things: power and authority. God has both, so He is sovereign. Calvinists instead believe that sovereignty is based on three things: power, authority, and intervention. Calvinists believe that if God does not ordain the actions of individuals in every aspect, then He is not sovereign. Arminians believe that since God created man in His image, it does not please Him to micromanage the affairs of man. It instead pleases God to give free will to man, to see if we will obey Him. God still has the ability to intervene when He desires to, and scripture indicates that He often does intervene. Arminians recognize that regardless of His level of intervention, God still has complete power and authority. He gets the last word. Man has the freedom to break God's law, but man does not have the capability of avoiding the consequences of breaking God's law. Thus God is sovereign.
So the key to understanding sovereignty is knowing that God does what He pleases.
It pleases God to give free will to man.