How does free will fit with the doctrine of predestination? The Calvinistic doctrine of predestination states that God has chosen who will live and who will die.  That Christians are unconditionally elected for salvation or damnation.

However, how does this fit the idea of Free Will?  If it's God that is choosing whether we turn to him or not, do we not really have free will?

(Please, Calvinistic views only.)
First of all, Calvinists do believe in free will. This point is often mis-understood by non-Calvinists; but the position that men don't actually have free will and control over their own choices is not Calvinism but hyper-Calvinism -- a deterministic view that goes far beyond that of it's namesake Calvin and the general constituency.

With that out of the way, the most important thing about the Calvinistic understanding of free will is that men are free to make choices, but only capable of making choices according to their nature. We can make any choice we like inside the scope of the kind of beings that we are but cannot make choices outside the scope of that nature or that defy it.

This is essentially the same sense in which we are limited physically. As humans we have finite power. The ability to levitate or spontaneously generate fire is reserved for the heroes of the Silver Screen. As humans we can affect our surroundings only insofar as we have the power to do so. But that power cannot change our nature. We can choose to eat healthy and exercise and not jump in front of moving trains -- and thus possibly extend our lives -- but we are still mortal and we cannot prevent our own eventual death. Whether through old age or sickness, we will die. Statistics are clear on this point. Our free will may allow us to live longer than if we had made other choices, but it will not allow us to not die at all. It is according to our nature.

In a similar sense, Calvinists believe that man has free will and is sovereign over the aspects of his life insofar as he has been granted these rights by God. However, we believe that man is, by nature, dead in sin. This means that it is not within the realm of possibility to "choose" salvation. A sick man may choose to take medicine and thus affect his own healing, but a dead man can do nothing to change his fate. This is the doctrine of total depravity and outside the scope of this question.

Calvinists believe that God is sovereign over salvation. That is his domain and for him alone to choose. The how that God chose to use, however, is quite interesting. First he extends us Grace. He sent his Son, he paid the price, he made a propitiation, he died for the ungodly, he ... he ... he. Then he holds all this out in front of us and he calls us. This call may be heard by many but only those who have been chosen are given the grace to respond to it, and through the hearing of the call are given faith and literally made into new beings -- resurrected from death to life -- born again as spiritual sons. This birth process isn't something we choose, it's something that is done for us. It's grace: unmerited favor. Something we could not do and would not choose to do had not God done something on our behalf first.

Calvinists may believe that God predestines people to be saved but they believe that man's limited scope means that we don't know who those people are. Furthermore as recipients of Grace we are called to share this news -- this free grace -- the Gospel, with all. We don't only preach to some. We preach it to all. Who God chooses to change into people that respond to the call is his business, but our business is inform everyone of the choice. Jesus called people to make choices, so do we.