It's mostly up to you... Start by finding a category that interests you (it must have less than a hundred sites including subcategories, and it's usualy better if there a a few listings)
Suitable categories always have a "become an editor" link. <add: or rather an icon - see this page>
From that you reach the application form - the three sample sites are for three sites you'd list in the category if you were the editor; so it's better that they aren't already listed.
Taking a look at our editing guidelines before applying can be helpful
I would like to apply for editing the Anarchist part of DMOZ or whatever the replacement is. The main page has a greyed-out icon with an onmouseover tooltip saying "please apply to edit a different category". From reading the editor application details it says more than one person can edit a section. Who do I need to approach in resolving this?
Society/Politics/Anarchism is simply too large to start out editing in. Please start with a subcategory, then apply for a larger area once you've gained some editing experience... But since the directory is technically shut, it's likely that there won't be time to review your application before the lights go out
“Dmoz is no more. It has ceased to be. It's expired and gone to meet its maker. This is a late mozzie. It's a stiff. Bereft of life, it rests in peace. If one hadn't nailed it to the perch, it would be pushing up the daisies. It's rung down the curtain and joined the choir invisible. This is an ex-mozzie.”
The editor community are currently working on a replacement for dmoz.