There are two ways people generally think of free will. The first way is to think of it roughly as the capacity to do what we want to do, regardless of where those wants come from. On this account, infringements on our free will come only from things like paralysis, imprisonment, poverty, and so forth: the freedom of our will is not diminished by the mere fact that our wants are the product of the collective influence of our genes and our environment. This kind of free will is called compatibilist free will. Specifying exactly what kinds of influences reduce this kind of free will is a subtle matter, and generates a lot of debate in legal systems no less than in philosophy departments, but it is certain that nearly everyone has this kind of free will to one degree or another.
The second way people generally think of free will is to view it as some kind of capacity to act contrary to the sum of all of the influences on us, without it being a random or purely probabilistic matter when or how we use that capacity. This kind of free will is called libertarian free will. Most philosophers think we do not have libertarian free will.
The main problem with libertarian free will is that the idea appears to be vacuous: when one says that an event is neither determined, nor random, nor a matter of pure probability, one appears to have exhausted all of the possible alternatives. The main things that can be said in favor of it are (1) that most people say that it feels to them as though they have it, and (2) that most people have a strong intuition that having it is a necessary precondition for having moral responsibility (which most people believe in).
For my own part, I do not think there is good reason to believe in libertarian free will. When people say that it feels to them as though they have it, I think this is an illusion that comes from not taking into account everything that actually goes into their decisions. If you think of literally everything that went into a decision you made—including the desire you may have had to demonstrate that you had libertarian free will by doing that which you normally would not—I believe you will find that those things jointly guaranteed that you would do what you did, unless somewhere in your decision-making process there was the equivalent of the roll of a die. If you do not see this, I suspect that when you try to rewind the clock, you unwittingly end up changing something in the original scenario, for instance by importing into your mind at that time the knowledge that you have right now. It is more difficult to see that our decisions work like this when we are in the process of making a decision, but that’s because up until the exact moment that we act, we continue to feel new influences, including the succession of our own thoughts, exerting themselves on us.
The question of whether libertarian free will is necessary for moral responsibility is a little bit more difficult. If we judge a person’s moral responsibility by whether or not his actions follow from his character—by whether or not he does what he really wants to do—then compatibilist free will is sufficient for moral responsibility. It is unclear whether this gives us everything we normally want from the notion of moral responsibility, but since the problem with libertarian free will is a strictly logical one, what we want from our notion of moral responsibility can’t have any bearing on whether or not we actually have libertarian free will.
I believe my judgments about all of this are borne out by the defenses of libertarian free will I have seen: they either amount to an appeal to mystery, which is always a losing proposition, or end up collapsing into something that is effectively a type of compatibilism, just hybridized with a little randomness. The consequences of there not being libertarian free will may be disturbing—I’m not really sure how one ought to feel about it—but it looks like it is a fact we must learn to live with either way.