Often, when students are asked to conduct a peer review in class, they think they're being asked to "fix" another person's spelling and grammar errors. While those kinds of errors do need attention, peer review generally takes place when the writing is still in a "draft" phase, meaning that the most useful help should focus on large-level issues, such as these:
After you've given the writer suggestions about how s/he can address these "big picture" issues, you may want to make another pass through the writing and provide feedback on low-order concerns, like grammar, spelling, missing words, punctuation, and MLA style. If the paper is going to need a lot of major revision, though, there may be no need for you to point out a misplaced comma, since that entire sentence may end up getting re-written.
Read like a reader, not like you're grading the writing.
During peer review, it's not your job to "catch" every mistake, nor is it your job to fill the margins with comments about every possible issue you find. Instead, try to read the writing from the perspective of an attentive, engaged reader--someone who is genuinely interested in what the writer has to say and wants the ideas to be clear, supported, and purposeful. From this perspective, it should be easy for you to spot (and provide feedback about) sections in the writing that are hard for you to follow, that seem to detour from the central goal of the text, or that seem to need further explanation or evidence.
When possible, provide suggestions for improvement.
Usually, writers are not intentionally unclear in their writing; most writers try to convey their ideas as clearly as they can. So, if you discover a place in the writinng that you think is unclear, a comment that only points out your confusion is much less helpful than a comment that offers a sentence or two about how the writer might revise that spot:
I get a little bit confused here. Do you mean that Kingsolver's concept of the "gangrenous leg" can be applied to the American education system? I think you're leaving out that connection right now, but the paragraph would make more sense if you added it at this point.
Even though your job is to read your classmate's work with an eye for improvement, you can frame your comments in ways that focus the writer on the potential of the writing rather than the problems. Providing suggestions for improvement is an easy way to help a writer feel positive about moving forward with their essay. Also, when you come across something you like in the writing, it's perfectly acceptable to point that out, too!
Value your opinion.
Sometimes, students doubt their ability to provide good feedback to their classmates. But, you can be sure that places that YOU find confusing will probably seem confusing to other readers, too, and that places where YOU feel unengaged by the writing will also seem insignificant to other readers. If you're willing to point those areas out, the writer can at least consider making revisions to improve clarity, purpose, etc. Trust your instincts.