How Does Story Editing Work?

I know what story editors are supposed to do, but how do they edit the story? Do they get the story pass and just fix the errors, or do they point out the mistakes to the author and they fix it?

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[An Alternate Universe]

There are a whole bunch of different forms of editing. A brief summary, then I'll go a little more in depth. First, editors job is not to fix the author's mistakes, that's on the authors. Editors only job is to highlight areas of interest and what could be done to fix them. Whether or not the author accepts these edits, or rejects them is on the author. This is true in the world of professional editing and it is true here, the only difference being that professional editors are... well, professionals and experts, so their advice is generally to be heeded, also, your work might not get published if it's not cleared by an editor. Now for the types of editing:
Developmental Editing (dev.editing)
The editor makes broad-stroke changes. At this level entire paragraphs, or even groups of paragraphs may be rewritten all in one go. Dev.editors are in a murky grey area where they are almost a contributor to the work, but are still firmly rooted in the world of editing. At this level common things to correct are awkward sentence structure, authorial quirks, dramatic abnormalities, delinquent tone (ex. comedic tone in a story rooted in a non-comedy, when it doesn't serve a dramatic purpose) and general weirdness in the story.
While I am doing dev.editing I also look for small mistakes, like having a double-space, or misspelt words. This isn't my primary goal, but if I notice something I'll correct it.
Dev.edits can change the entire story at times, simply by their nature.
Generally I end up doing two or three rounds of dev.editing, I can assure you that one round is not sufficient.
Copyediting (CE)
This is the slowest type of work, but it requires the least input on the author's behalf and is rather objective (you can love or hate what you're editing, by the end of this you will be intimately acquainted with it and not care either way). Each word needs to be reviewed, each sentence read aloud and each paragraph taken in the larger context. Punctuation, spelling, grammar, fact checking and detail checking take place at this level. When doing CE work very little occurs in the way of major changes, and the most that should change would be a capital, a space and a punctuation mark would be deleted and replaced with a comma and a space.
This is painstaking and it helps if your editor can zoom in and if the text has lots of spacing between lines (it can be corrected to a more compact form later). It takes several hours for a few thousand words and is often done over several days (at least in my experience).
Proofreading
This is not just skimming the document; it's making sure that the document is ready to be published. There are two forms of this; the first is to check over the document in its draft form and read for errors and to take it holistically, that is, to ensure that it makes sense. The second is to check the final document in its publishing format, that is, to check tagging is correctly applied, pictures are where they are meant to be, etc. This shouldn't take as long as either copyediting or dev.editing, but it can still take a while.

Hope this answers your question.