This is a very tricky subject that delves deeper into philosophy than I'm really comfortable with. I enjoyed my introduction to philosophy class in college, and the professor clearly enjoyed debating with me, but I got turned off the topic when I started encountering people claiming Aquinas' Five Ways as proof that their god must exist since the concept is logically sound. The problem is that logically sound arguments don't always translate to physical evidence; just ask the Nineteenth Century scientists trying to verify cosmic ether.
So let's begin at the beginning. What is "free will?" The definition provided by that link gives us a good place to begin: “Free Will” is a philosophical term of art for a particular sort of capacity of rational agents to choose a course of action from among various alternatives. Summarized even further, it's our ability to make choices free from interference.
"Determinism" then is the polar opposite of free will. Again citing this link, Determinism is the philosophical idea that every event or state of affairs, including every human decision and action, is the inevitable and necessary consequence of antecedent states of affairs. Free will is impossible because we're necessarily bound by the choices and events that came before each choice.
The jury is out on this topic, with new evidence coming to light tipping the scales back and forth. The latest findings in neuroscience suggests that free will is just an illusion born out of biochemistry.
I still like the idea that I have free will, that the agency behind my choices aren't strictly determined by biochemistry and causality. I can choose to set myself on a different path if I'm determined to do so. However, I can't dispute the evidence leading to the answer that it is all biochemistry and causality.
Does that have to be the end of the conversation? Our brains are based on complex neurology that produces an emergent property we call consciousness. The foundation of our brains is really very simple and its components can be found in any creature with neurons. Can it be that free will is also an emergent property from the complex interaction between biochemistry and causality? Biology may make us prone to certain actions with causality boxing us into a series of choices, but we've seen how quantum effects collapse in the presence of an observer. We behave differently when we know we're being observed or tested than if we think we're acting on our own.
I think it's a hideously complex topic, and until we gain better understanding of the nature of consciousness itself we won't be able to answer this question with any accuracy. For now I'm going to continue to hold on to the comforting illusion of free will as a justification for holding myself accountable for my actions.