Determinism is the philosophical position that every event is caused, and is completely explained by, the preceding events. The causes, in turn, are events that were caused by earlier events. Nothing happens without there first being some event sufficient to cause it. This is a natural conclusion following from Newton's observations on the natural laws that (seem to) govern all aspects of our universe.
In most cases, we believe that an accurate description of an event includes all causes. If we want to know why the eight ball went in the corner pocket, we look at the exact force of the cue hitting the ball at a specific angle, we talk about the friction between the table and the ball, and the transfer of force between balls and the angles at which balls bounce off of the sides of the table. We generally talk about gravity and inertia and air resistance, but if the ball goes into the pocket because the ball wants to go into the pocket, well, we'd better include that too. The explanation of what happened includes all the causes, and an explanation had better include all the causes, whether we like them or not, or it simply isn't a correct explanation. Determinism is simply the blanket affirmation that this method of explaining things is the correct method, in every context.
The biggest problem with this idea is that it seems incompatible with free will. There are some sorts of 'mental events,'* which include our feelings, wants, and decisions. These events must surely be caused in the same way as physical events; you want something because you've learned that it is good, you love someone because of the way they are (and the way you are). The wants and likes and knowings come from things that have happened to you, either inside your head or out. You are the way you are because of the way other things were. An explanation of what you do is composed of an exhaustive list of causes.
The alternative may be worse; if things (including mental events) aren't caused, they are uncaused, random. This 'random' is not random in the sense that heads or tails is random; A coin flip is caused, and with enough data (and computing power), we could indeed tell if a specific coin flip is going to come up heads or tails. (This is rocket science.) The random we are talking about is a nothing-BANG-something type of random, which is quite literally impossible to predict. Needless to say, random, unpredictable, and spontaneous thoughts are not the type of 'free will' we want causing our actions.
At the moment determinism seems to fit in very well with what we know about the universe on the marco level, but quantum mechanics seems to support the idea of random events, at least in a sense. Whether this is a boon for free will is questionable, as being probabilistic is a rather unsatisfactory cross between the worst of determinism and chaos: perhaps on a quantum level we neither choose what to do nor can we predict what we might do.
* Determinism does not require that you believe in materialism, although I think most determinists are materialists. You may replace this with 'chemical events', if you like.
Regarding the argument that determinism frees us from responsibility: If we believe in determinism, then we should believe that we have responsibility for our actions. Admittedly, 'responsibility' is an abstract term, but there is nothing to be gained by saying that we don't have it. In the worst case, responsibility is simply a specific "program" we run to determine the most effective reaction to certain events.
If Determinism is true, then what we believe is not under our control (at least not in the way most of us would like to believe), but what we do is still decided by what we believe. You may not be able to decide, freely, that you should act good or bad, but you are still an agent who does make decisions, and these decisions are affected by the predicted results of the decisions. Of course, if you decide, for any reason, that you are not responsible for your actions, that will change the way you behave, whether or not you are correct. It may not, however, change the way other people view you, and will not exempt you from any of the systems that are in place to deal with people to behave in ways we don't like.
Punishment, in the sense of 'hurting' a person in order to change their behaviour, is still valid under a deterministic system. Perhaps even more valid. Revenge is not valid under determinism, but then, its not valid under any system.