In a move that strikes me as completely identical to Adventure Mode in Diablo 3, there are a variety of quests to complete in each zone. Complete four of them for the zone's Emissary and you will receive a bonus chest! Sound familiar? Each quest also has a reward upon completion - gold, an item that grants artifact power, or a piece of gear. You'll want to stay on top of this, as quests rotate fairly often.
Enemies that are related to quests in a given area are also able to be tagged by multiple characters, which is a nice change. You'll still see people try to group up as many mobs as they can for fear of other players ‘stealing' their mobs, but in most cases, it's safe to share.
These haven't been released just yet, but you can think of this as the WoW version of Greater Rifts in Diablo 3. Completing a Mythic dungeon will grant you a keystone that will allow you take on a Mythic+ version of the same dungeon. These dungeons are not only quite difficult, but they will receive some kind of modifier to up the challenge, in addition to simply scaling up the HP and damage of the enemies. Completing a Mythic+ dungeon will level up your keystone and allow you to take on an even more difficult version.
In addition to all of the above, Blizzard has stepped up their story presentation quite a bit. There is voiceover just about everywhere, which helps the world feel much more alive. Heck, the very first thing you'll do in this expansion is join a 25-man raid that is absolutely packed with story. The new zones are among the most beautiful I've seen. Class identity is at an all-time high. Because of the campaigns and order halls, I actually feel like the class I chose matters in the grand scheme of things.
While World Quests give players more reason to work together in the open world areas, much of the structure of Legion is similar to what's come before. Fortunately, WoW has always had a healthy mix of single-player and group content for all skill levels, and that hasn't changed.
All in all, if you bounced on WoW long ago, now's as good a time as ever to come back. It doesn't hurt that each copy of Legion comes with a free level 100 character boost!