Supervising the editing. Some directors will sit with the editor most of the time and tell them what shots to use, others will let the editor work for a few days or weeks on their own and then come in to see what the editor has done, making suggestions to improve it.

Directors will pick out takes, the nicer ones will give the editor time codes.

Beyond that it's really up to the director, as Mark mentioned. This may be a result of being an editor as a day job, but I tend to back off and let the editor do his stuff, then come review it and see if I want to make any changes. I mean, there was a reason I hired an editor. If I wanted to control every aspect of editing I would have edited it myself.

I could be wrong but in most cases an editor starts with a shot list, and a list of which takes were good so they know to a pretty decent extent what the director is looking for already.

After that it's give and take on what the director didn't consider (after all they are often too close to the material and may lose some objectivity) and things the editor perhaps didn't think of (after all they are new to the material and may not entirely get some aspect) and a lot of fine-tuning and occasional patchwork (that is sounds that link scenes, or sound cues that introduce elements that otherwise seem to appear really suddenly, or even a cut showing a characters location to help smooth out the visual geography).

RJSchwarz
San Diego, CA

It's also not unheard of for a director to edit their own film. This is especially common in short and independent work, but the Coen Brothers, for instance, edit their own films under the pseudonym Roderick Jaynes. On the other end of the spectrum, some directors just hand the footage to their editor and walk away. Walter Murch, for instance, more or less completely re-invented The Conversation after Coppola shot it.

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Andrew Gingerich
Exploding Goldfish Films
Check out my blog at http://www.exgfilms.com
and my reel at http://portfolio.exgfilms.com