Compatibilism, in the context of philosophy, is the belief that humans can have both free will and determined actions. To understand compatibilism, it is important to first understand the two arguments it brings together; determinism andfree will.
Free will is easy to understand; it is simply the idea that, as humans, we have the ability to make independent and free decisions.
Determinism however, is the philsophical argument that nothing can happen without being caused to happen. Just as a pen will not move unless something makes it move, determinists argue that people can never make a decision without that decision being influenced by their past experinces. Thus, according to this theory, all our actions are determined by our thought processes in making the decison, which are formed entirely by our pasts. Thus, if we knew enough about a persons past, character, and thought processes, we could predict all of their decisions. And if all of our decisions are determined, how can we ever act freely?
This is where compatibilism comes in; compatibilists like Hume (probably the most famous compatibilist philosopher) argue that just because our actions are predictable, does not mean they are not free. By challenging the defintion of freedom as we assume it to be, compatibilists argue we can exist freely within a determined universe. This is because, as Hume argues, freedom does not mean the ability to make decisions independently of our pasts, but the ability to make decisions without these decisions being forced upon us. Simply because our actions are predictable, because of our individual thought processes and experiences shaping the way we make decisions, does not mean that we are forced to make these decisions. Thus, compatilists argue, although our actions and decisions are determined, we still act freely.
Once you understand these themes the best way to answer a question on this topic would be to outline the concepts, then discuss how they link together through compatibilism. Then you can make your own decision on what to conclude in the essay; is is possible to argue all three sides! Remember when making your argument to discuss why your decision is a response to the other arguments, not just stating what you think. Don't forget this is a hard topic, so if you understand the concepts and can evaluate them in an exam, you'll do well.