I've refereed at least a dozen papers in my (short) career so far and I still find the process completely baffling. I'm wondering what is actually expected and what people tend to do...
I used to think that I had to check a paper's correctness, but now I think that the main point of my report is to help the editors decide whether they should accept or reject the paper. It's the author's job to make sure the paper is correct. This has few practical consequences:
- I'll agree to referee a paper even if I don't think I'll be able to understand everything in it, but I'll send it back as soon as possible (within a week or two) if I feel that I can't even tell whether it's interesting or not. In this case, I'll suggest some other names.
- I do summarize the main result, so that the editors know what it's about. I try to do this in half a page. If I'm not able to do this, it's a bad sign. It has happended that I couldn't tell what the author was trying to do.
- I give a short opinion about acceptance or rejection, it's usually not hard to decide.
- After that, I include a list of the errors and typos which I found when reading the paper.
Because of the way I do this, I'm usually partial to a good introduction.
Here are a few things which have displeased me (as a referee) about the process:
- I recommended that the paper be rejected because even though it's close to my subject, I couldn't understand anything, but the editors said that they had sollicited the paper and they felt obliged to accept it.
- I made a four-page list of errors and suggested corrections, which the author ignored, and after a couple of back-and-forth, the editors asked could they please just accept the paper as it was.
- The editor was a friend of the author and felt I should not have been so harsh.
I agree with Richard Borcherd's Borcherds' recommandation that one should write assuming that the author will find out the referee's name. But recall that the author will see your report in a quite different light. I once started a report with something to the effect that the paper was very good and useful and bound to become a reference. A few weeks later, I was talking to the author who said the referee had written a "quite negative report".
This is the way I do things, but I have to say that receiving a six-page report where the referee has checked all my computations (including the signs!) is quite wonderful.
I think it would indeed be helpful if each journal was more precise in saying what they expect of the referee. Conversely, as a referee, I like to know the outcome (some journals received my report and that was the last I heard from them). Sending a reprint of the paper once it has appeared is a nice gesture.
My last comment is a reiteration of the fact that a timely report is important and useful. If you can't do it in two or three months, you've got to ask yourself if you are going to do it at all...