Think of the Grand Alliances like the allies during WWII. You've got your allies that generally get on with each other and those that historically haven't (Britain and France) and you've also got some nasties in there like Russia who you know are just as morally bad (if not worse in some ways) than the one you're fighting against; but you don't want them as an enemy and you'd rather have them as an ally against the closer threat.
Grand Alliance Order are those who have remained within Sigmars Grand Alliance (which used to include both Destruction and Death) against the forces of Chaos. It is basically the force that will fight against chaos against all odds and at the same time have retained ties to Sigmar and his Stormcast.
They are also generally seen as civilized peoples who build cities and cultures of their own.
Grand Alliance Death are those forces allied (controlled) by Nagash and are generally rather undead and decaying. The Rise of the Stormcast and Sigmar stealing souls that Nagahs feels should be rightfully his; caused Nagash to break his alliance and - as the Age of Sigmar has pushed chaos back a considerable way - Nagash has now broken ties to Sigmar and the alliance and wages war upon them as well as upon Chaos. Nagash's forces can build settlements and civilizations but they are undead and hold allegiance to Nagash
Grand Alliance Destruction are those forces that don't build civilizations nor settlements in the same way. They are the wild untamed people who are generally nomadic and who will raid and harry and steal from others. Generally those who like a good fight without any rules and who are just in it for the action.
They are not stupid, they are just not your typical civilized peoples. Think of them a bit like Conan the Barbarian - they can perfectly operate within a civilized settlement, but in the end they chafe at the bit; go a little nuts and just have
ot strike out back to the wilds and fight.
They have also broken their formal Alliance to Sigmar and his peoples.
Grand Alliance Chaos is all those forces tainted by the ruinous powers of Chaos. Whilst that originally was just the four gods it now also includes the fifth god - the Great Horned Rat (hence Skaven are here too now). They've also got forces like the Beasts of Chaos who are more a force of Chaos than those who are allied to specific Chaos Gods from within the Chaos Realm.
Within Order specifically you have got wildcards; eg the Daughters of Khaine who are almost as close to vampires as you can get without being undead. They are bloody and violent and also allies who can turn on you in battle without remorse. However they are fanatical in their hatred of Chaos (esp Slaanesh) and will fight hard against such forces. They also build and found settlements and Temples; why your local city likely has one of their fighting-pit temples where you can go to watch the matches. That the blood shed within goes to feed Khaine's power (well Morathi) is a side effect most commoners know nothing of.
There are others too. I think this panel from the Roll Models comic sums it up rather nicely.