It depends upon what you need to edit, but here's the basic process:
Copy the URL of the existing story.
Start a new submission. For the title, use the same title as the original ( or as much as will fit ) + a notation such as *EDIT*.
( Use *DEL* or *DELETE* or something if you're deleting a story, rather than editing )
The "Notes" section is the most important part of the submission. This is where you'll describe what you want done to the existing story. Paste the URL of the original story here, because that helps define exactly what you want edited, and reduces the chance of human error.
Then, describe what you want done. New title, new description, new keywords, category change, etc.
The rest of the submission form can be filled in with placeholders, or if you're changing one of those things, you can put the new thing there. The "Notes" section is still where you should define the changes, though.
If you're editing the story text, you need to upload/paste the WHOLE edited story or chapter into the "story text" section. Not just the parts you edited. You need to send in a complete, edited manuscript, which will replace the existing one.
If you're only editing some other element such as title, and not modifying the story text, then copy-paste what you put in the "Notes" section into the "Story Text" section. You need something there, and duplicating the notes is a sure way to make sure it gets noticed.
Once that's done, preview and submit as usual.
Laurel has said that edits have a lower priority than new stories, but that seems to vary depending upon her workload. So, expect it to take longer than the standard 2 days stories take to post.
And keep in mind that the site is cached. Your changes, once processed, won't appear on the public site until 24-48 hours afterward. The way you know the edit has been processed is that the new, edited submission you made vanishes from your private author submission list.
Not all of the changes may show up at the same time, either. I've seen page 1's changes go live, while changes to page 3 take another couple of hours to show up. Once the modification starts going live, it's never been more than a couple of hours before everything catches up, though.
While I'm at it, the editing process does not affect existing votes, views, comments, favorites or anything else. It will not appear on the new story lists again.
If you want to start fresh with a story because you put it in the wrong category and it got trashed or you've so significantly modified the original that it's hardly the same story any longer, you need to request a delete for the original, and then post the new version as a new submission once the delete processes.
Unless you've significantly modified the original, you'll probably want to mention in a note at the beginning that it's a repost. There are enough readers out there who will leave nasty comments and 1 votes for an unmarked repost that it's worth it to take the time.