Battleborn

Advanced difficulty is a more difficult mode that you can play in Private Story lobbies.
Devs have said they plan to patch in Advanced Public Story matchmaking in the future.
I don't know the full extent of the changes this mode brings over but at the very least enemies do much more damage and the type of enemies that spawn isn't the same.
It's advisable to play with other people, going in solo can be borderline impossible for "defense" segments.

Progress can be checked at "COMMAND" section on the main menu (third option) and then pressing twice to get to the "CAREER" tab. It has two columns, one for playing just Advanced difficulty and another for playing with Hardcore on. You can probably mix and match between the two but I'm not 100% certain.

Here is a mini-guide I've written up for each stage.

The Algorithm
This stage is pretty straight forward, just kill things and advance through. Maybe have one person pick Miko so they can heal people.

The Void's Edge
Much like the previous stage it's pretty straight forward. The second half of the stage you'll have to guard the wolf bot but as long as you stay on top of the enemies spawn points you should be fine.

The Renegade
First defense segment build a Fire/Temporal Trap by the stairs next to the wide open area. Then build a rocket turret. Do your best to defend against the enemies.
Second wave watch the other side of the objective as the little exploding bugs will spawn nearly right next to the object you need to defend.

Second defense objective is much easier. You don't need to build anything the first wave. Second wave maybe build one of the Fire Traps by the entrance of the base, Maybe a Trap behind the objective. A a rocket turret or three.

Third defense shouldn't be that hard only as long as you stay diligent in destroying the enemy spawn "Anchors". On Normal difficulty first they'll spawn on one of the two outside platforms, then the next wave they'll move to the other one and then it gets to the free for all and they come in from the roof and all that. On Advanced they'll start on the roof at the very first wave, so have at least one person up there trying to distract them from breaking in and taking out those roof spawns which sometimes will spawn another spawner almost immediately after it's destroyed.
I suggest all rocket turrets inside and then two Fire Traps, Temporal is fine too or you can even mix and match.

The Archive
This one shouldn't be all that hard, you have the first escort defense, just stick close to the objective as it moves and do your best to clear out the enemies. If you were to fail this would be the segment it would happen.
Next you have to protect the minions as they get to the bigger bot you were escorting. I don't think you can fail this segment but still don't waste time and defend the minions to the best of your ability.
Next you have to escort the bigger robot again, should be no problem at this point.
Finally you should be at the boss and while none of the bosses pose much of a threat this is probably the easiest one.

The Sentinel
No defense section so this should be easy. Just be careful during stage hazard sections. You do three boss fights but they're just giant damage sponges.

The Experiment
Second hardest stage. You have to defend the same objective twice, the second time is against a mini-boss. More info to come.

The Saboteur
!IS GOING TO GET PATCHED BY DEVS TO BE MADE EASIER!
Currently the hardest stage in the game... With a group.
If you're capable of managing groups of enemies alone then solo is probably the best way to go. Any character with a stun/slow/AOE attack should be good.
Oscar Mike is good because he has both stun and slow AOE attacks.
Rath is doable.
I personally used Marquis, I'll detail his build below in a spoiler.

If you go the Marquis route, don't play him like a sniper, play him like you would Oscar Mike minus the part where he runs away like a little girl. Marquis doesn't have that problem, he's too busy slowing and blowing away everything in front of him, basically spam both your abilities as much as possible. Get some of those Goldeneye tier pistol shots in while waiting on cooldowns. The accuracy on it is wonky but a single shot does so much damage that the majority of shots you'll land makeup for the ones you'll miss.
I'll finish the rest of this Marquis walkthrough later.

Rough draft of the actual stage walkthrough:
Basically build the Thumper Turrets during each defense segment and then basically whatever traps you want everywhere else. The smaller minions that look like spiders are very deadly on the third and final segment. Fire Traps on the top of the two sets of stairs could be good for damaging and distracting them. Remember to replace things as they get destroyed, throw down a slow or AOE in the little doorway by the defense objective while you run to replace them.

The Heliophage
I've managed to pull off a solo on this stage with Oscar Mike.
First part of the stage you start with 24 minutes. Do your best to clear out sections of enemies and advance to the next location on your map as soon as a new objective pops up. You'll arrive at a objective that needs to be destroyed, clear out all the enemies as fast as you can. Once all enemies are cleared you'll have to do the same thing again on the opposite side of where you were, a small bridge connects the two areas. Once cleared you'll have to go back to the starting area of the stage. I got back with about twenty seconds left on the clock, the switch you have to stand on to get to the next area takes about five seconds to activate.
Now the real part of the stage takes place on a bunch of floating islands and you have to face off against pretty much every stage final boss you've faced before minus Isic and The Sentinel. You'll probably die a few times, that's okay as every boss you beat will drop a extra life. Use your cloak when you're low on shield, stay away from bosses and clear out minions first. This stage took me about a hour and five minutes to complete, it would probably gone a lot faster with more people but that's your call, just letting everyone know it's possible.