* Notification emails that are very time-specific in their relevance. "Reminder: Coffee with Chuck at 3:25 today" or "Thank you for your reservation for a party of 4 at Alice's Restaurant at 7:30 tonight." You care about these a lot, until you have coffee with Chuck and abruptly stop caring.
* Things that are nice to have pass through your consciousness at some point. Amazon shipping confirmation emails, GitHub comment notifications, etc. Interestingly, most of these are mostly noise; when I get an Amazon confirmation email, the part I care about could fit in a Tweet: "Your Dromiceiomimus Plushie has shipped via USPS, with tracking number 54989086, and will arrive between Tuesday and Friday".
* Mailing lists. I have Gmail set up to tag all these and skip my inbox, which seems to be about right. The weird thing about mailing list emails is that almost all of them are explicitly marked as such -- there are several unofficial MIME headers for this -- and yet the default behavior for every email client I know if is to simply stick them in the inbox like any other email.
* Crap I don't care about. "Save up to 25% on useless ornamental doilies!!!1". The only part of this that I care about is the Unsubscribe link. I believe Gmail can auto-unsubscribe you if you mark these as spam.
Of course, while it's fine to break it down like this, the trouble starts when you start trying to actually extract useful information from emails so you can present users with a better view of their inbox. It's officially a Hard Problem, though like a lot of hard problems, you can get something useful with a lot less effort than getting something perfect.
What would you like from your inbox? For obvious reasons, I'm very interested. :-)