That being said, my editing process is a multipass approach. The numbering is for convenience of discussion more than a reflection of either order of priority or order of process.
First is the line edit. This is simply reading through the manuscript for typos, wordos, dropped words, duplicates, and so forth. Nothing ever goes out the door without this edit, though sometimes it has been the only edit I've been able to accomplish on a given piece.
Secondly, and often done simultaneously with the line edit, is the style edit. This is reading through the manuscript for issues of prosody, rhythm, rhymes (accidental or intentional), alliteration (accidental or intentional), echoes, clunks, infelicitous turns of phrase, and so forth.
For both of these editing layers, a read-aloud is extremely helpful. Here's where one of my idiosyncracies comes into play. I pretty much can't manage a read-aloud by myself. I am virtually shameless, as many of you know, but reading aloud to myself in an empty room feels too much like being the crazy uncle in the attic. So I really like to read to someone when I can. (This isn't terribly practical on a novel, sadly, at least not for me.)
Third is a structural edit. This has to do with things like sentence order, scene order and so forth. A fairly common error in writing, for example, both mine and other people's, is reversing cause an effect. "He turned as a shot rang out" is backward, to use a very simple case. This happens because as I'm writing I arrive a bit of plot or action, then rationalize it into the story, but don't realize I'm doing it. It can happen at the line level, the paragraph level, even the scene level. Occasionally even the chapter level.
Fourth, and also often done simultaneously with the structural edit, is the plot edit. This is making sure the events and characters progress in the order which the story calls for. Linear is nice if you're telling a linear story, but if you're telling a non-linear story, then the order of the reveals is far more critical and tricky.
Other passes (or at least watch-fors within passes) include a continuity edit, reading for character development and arc, and a high level readablity check on the story as a whole.
I strongly recommend against having a checklist approach to these processes. I do multiple editing passes when time and energy, but in reality I'm reading at most of these levels most of the time, with a shifting focus with each pass. Much of my journey as a writer has been to develop a holistic editing approach that encompasses these layers, along with more granular issues (handling gender perspectives, for example). Right now I'm shifting focus with the goal of achieving a more conscious competence, dialing up specific aspects of editing to work on them.
I don't have good practical advice here, just some descriptive narrative. This is my own weakness as a working pro, which I need to advance beyond to improve my craft and professionalism.
Comments? Your own observations and experience? Suggestions for me?
ETA: As I should have pointed out in the first place, the brilliant yet talented kenrand has written what may be the best book on editing process ever, The 10% Solution [ Amazon ]. I highly recommend this book, which in addition to a much more detailed high level overview than this post can encompass, also has some excellent nuts and bolts suggestions for the editing process.