Despite having the most powerful AI in the galaxy connected directly to your brain with an instantaneous connection which spans lightyears, you can only check your email by actually logging on to a computer aboard the Tempest. There are two you can use, one in your quarters, and a much more convenient one on the bridge.
The email organization seems nonsensical at first, but it's loosely grouped by sender (so all of Cora's emails will tend to clump together, even if you received bunches of other emails inbetween). Unread emails always rise to the top, with little blue exclamation points, and emails that start or forward quest-lines will have little hexagon markers next to them.
Some quests (particularly long multi-parters like the the loyalty quests or Path of Hero) require you to check your email, but yo uprobably should anyway. It provides epilogues to a lot of the lesser choices you make, and few of them are just hilarious on their own.
You can also archive old emails to get them out of your inbox.