"Kill" makes an item be "[dead]", i.e. hidden from users who don't have the "showdead" setting turned on in their profile. A typical case is killing duplicate posts when they don't have an ongoing discussion. "Blast" is like "kill", but also bans the user. A typical case is banning trolls. "Nuke" is like "blast", but also bans the site. That is for spammers.

Oddly enough, no one asked about the most important by far of all these links, perhaps because it also has the blandest name: "edit". That goes to a page where we can edit an item's title or url, or put some moderation flag on a story.

I'm not just speaking for myself here, but for everyone who's done it.