Gateway is a part of the Neverwinter website that allows players to access their characters, without logging into the game client, in other words, from their browser or "on the go."
The address is: https://gateway.playneverwinter.com/
In Gateway, your character sheet(s) can be viewed to a light degree (stats and equipment, not powers/feats/companions). An update is intended to allow greater inventory management through Gateway. Edit: This is live with Fury of the Feywild. You may now sell inventory for gold/copper/silver. And buy profession assets and resources.
You may read your mail, but not take items.
You may manage your professions, which is a very useful feature.
On the Guild view you can check the general overview as well as the News, planned Guild events, Member list and the Information site of your guild but at the moment it is no possible to do anything but check this information.
You may access the Auction House (players trade with other players, using Astral Diamonds as currency. This is useful because you can watch your bids closely while not playing Neverwinter (on the go, or while doing something else on the computer).