If I refereed a paper at one journal (and it was rejected) should I referee it again when it's sent to another journal? A couple months ago I refereed a paper for one journal. Due to a number of factors, I recommended that it not be published in that particular journal, and gave the author a bunch of comments on how they could easily improve the manuscript. The editor of that journal followed my recommendation (and perhaps the recommendation of other referees I am unaware of) and did not accept the publication.

This morning I received another review request, this time from a different journal but for the same manuscript. I've never had this particular situation occur before, and am wondering what the expected thing for me to do is. Should I review the manuscript again as if I hadn't before, should I let the editor know first and see what they say, or what?
There are no problems associated with you reviewing a paper again, regardless if it resubmitted to the same or a different journal. Since you have seen the paper earlier and know its earlier problems you can better judge how it has improved and to what extent it is now publishable. You should definitely let the editor know that you have been involved in the process of this paper earlier even if he/she has not indicated the paper has been rejected somewhere before. That information provides the editor with a better perspective of the development of the paper and can make the decision to accept or reject easier since the willingness or capability of the author(s) to improve the manuscript is set in perspective.

You can always contact the editor and state that you are willing to take on the review but felt it was necessary to convey the information. I do not see it as necessary but it is of course a nice gesture.