Just wrote a post on how to review a paper, but I realized that there's another step to review, which is how to deal with revisions. Anyway, here's a small follow-up on how to re-review a paper, which is when the authors revise the manuscript and resubmit for you to look at and approve.
1. If you now find something problematic from the original submission that made it into the revision, well, too bad. In my opinion, you are not allowed two bites at the apple. I realize that this is sort of "bad" in the sense that the manuscript may be published with a flaw, and reasonable people could disagree on this point. But this is my policy, because it's just absolutely deflating to deal with this as an author. I should say that it is pretty rare for something really critical to only surface at the second round. When I've gotten these second round criticisms, they are typically just for inconsequential stuff that the reviewer is trying to trump up because they are a jerk.
2. Keep it short. Like, two lines or so.
3. Don't get mad if they didn't do everything you asked. I had one reviewer for a paper who clearly didn't like our work ask us repeatedly for more experiments and by the end basically just said something to the effect of "I just feel like they should have done an experiment." Not cool.
At this stage, the editors have probably basically signed off on it (though not always, we all know horror stories), so I don't think it's worth getting too involved.
Another, somewhat related issue: how to review a paper that you've already reviewed for another journal. Only been in that situation a few times myself, and it's always weird. I guess you can just recycle your previous review, unless the paper has changed somewhat. Or just decline to review the paper (seen that as well). Better thing is to just try and get the paper accepted at the first journal so that you don't ever have to see it again.