When someone critiques my work, the most helpful thing anyone can do for me is to comment on my plot and characterization. These two things have always been my weak point. Of course, I'll take anything anyone deigns to give me!
Now, a good critique. That varies from person to person. I've found the best critiques to be in line-by-lines, because when you do a line-by-line, you're more likely to catch problems (unless you skim). Many people balk from these, however, because for one, line-by-lines are time-consuming. And two, they're often punished, not rewarded. I'm sure many of us here have gotten bitten in the ass for taking time to critique someone's work.
Overall, the best critiques will come from those who read a work carefully--not so easy on the computer screen, I know--and are willing to respond to any and all problems they see.
When I'm giving people advice on how to write good critiques, I always tell them approach the piece like an acquisitions editor first, a writer second, a reader third, and, if applicable, as a friend fourth. Look for plot holes, awkward phrasing, annoying or unbelievable characters, etc. Look for repetition, echoes. See if the story pulls you in as a reader, if it appeals. See if the writing is technically correct.
I also tell people to remember that when they critique someone's work, their critique is tinged with their own personal style, so they should check to see if that thing they're commenting on is really not good for the piece or if it's just a harmless stylistic tic. (Stylistic tics that are any good are pretty rare, though.)
Tehuti, I agree with you. Specifics are really helpful in a critique. Even if they're relatively vague specifics. For example: "Something about your writing just grates on my nerves. I think it's because your character's an annoying little git."