Hard determinism and libertarianism agree that determinism and moral responsibility are free will are incompatible--that if we are determined in our behavior, we cannot be legitimately held responsible for what we do. They disagree, however, about whether we are determined. Hard determinists say we are, libertarians say we are not. Taylor is essentially a hard determinist. In his paper he first gets clear on our conceptions of determinism and moral responsibility, then argues that these are in conflict with one another, and so one must be incorrect, and finally argues that it is our thoughts about moral responsibility that must go, because on reflection the idea that we are determined is the stronger of the two.
This article does a nice job of showing a typical philosophical activity: it starts by getting clearer on thoughts about ourselves that we all hold more-or-less implicitly. It makes them explicit and shows that once they are articulated, we can see that they are in conflict. It then considers whether there is some way of reconciling them, and if not, which of them is the more likely to be true. So, with this template in mind, let's look at Taylor's paper.
1) Definition of Determinism pp.437-439
Here he gives us an account of the idea of determinism that we presuppose in our day-to-day lives, and some of the reasons we presuppose it. We think that the state of the universe is fixed at every moment, and that its character is fixed by its state the moment before. Since we are part of the physical universe, this must be true of us as well. On 438-9 he tells us that this applies to human behavior as well, and gives us some details about how it does.
2) Our Ideas about Moral Responsibility pp.439-42
Here he gives us an account of the kinds of experience that makes us believe that we are morally responsible for some of our actions. He points to two things: first, that we sometimes deliberate about what to do, and second that whether or not we deliberate, it seems as if sometimes what we do is "up to us" and not settled ahead of time. He describes each of these in some detail. On p.440 he gives four conditions of deliberation, and on pp.440-2 he talks about the notion that what I do is "up to me." Like Campbell, he thinks this has a strong sense, and that it has to do with my really controlling the outcome of events in a very strong way.
Deliberation and the feeling that what we do is sometimes up to us are the two data (the singular is datum) supporting the legitimacy of moral responsibility. On p.441 he says that we now need to consider whether these two data are compatible with determinism, and if not, whether we should reject determinism or say that the data are mistaken or illusory. He also notes (p. 442) that the doctrine that we are responsible is of much more practical importance to most people most of the time than the doctrine of determinism, so unless we have strong reasons favoring determinism, we are more likely to accept the doctrine of responsibility.
3) The Two Data are Incompatible with Determinism. pp.442-445
He first of all distinguishes causal necessity from logical necessity. Here he is basically making Campbell's move of rejecting weak readings of "could have done differently." Next he gives a definition of "freedom" which reinforces this idea. After that he considers soft determinism, which claims to reconcile freedom and determinism, and says that this does not solve the problem by just "camouflages" it (443). What does this mean? He gives a very lucid argument against soft determinism on pp. 443-445, make sure you understand it.
4) Determinism is the Better View pp.445-449
First (pp.445-7) he offers the argument we have already discussed that indeterminism--that is, libertarianism--is unintelligible. At the very least, he argues, it does not provide the sort of freedom we want for moral responsibility. On pp.446-7 he thus argues that the two data are not compatible with either determinism or indeterminism. Finally, on pp.448-9 he argues that the two data really assume an extraordinary and indefensible view of the world--especially of agency and causation. Because of all of this, he argues that determinism has the most going for it, and that in any case the two data have little going for them.