I'm somewhat confused on how all these different campaign settings fit together. Is each campaign setting entirely different, each with its own cosmologies as well, or are there multiple worlds on the material plane? 

Depends on how you spin it.  Some campaigns are set in the same world (Torril for the Forgotten Realms which includes the Sword Coast, the Dalelands, etc).  Some are set in different worlds on an alternate Material Plane (say, Wildemount in Exandria from Critical Role or Grayhawk of the original D&D materials).  They can easily be connected by the Astral Sea... or portals... or whatever sparks your imagination.  I'm sure there are very detailed explanations out there (which include the Hells, the Abyss, the Planes of Earth, Air Fire and Water, Elyssium, Valhalla, etc...) but mainly use what makes sense to you and if you ever need to connect them, then look into these kinds of places and the Astral Sea.  The Gith and the Illithids will thank you.

I would say originally intended they are distinct universes unrelated to each other.  I would go further and say every single DM has their own universe.

Naturally any individual DM can grab any piece of any campaign and drag it into their universe.  Certainly back in the day I pulled the Template of Elemental Evil from Greyhawk and plopped it into my Faerun campaign.  And with that said they can make each campaign setting a different world with a shared Cosmology.

From the introduction pages of the PHB:

Worlds of Adventure

The many worlds of the Dungeons & Dragons game are places of magic and monsters, of brave warriors and spectacular adventures. They begin with a foundation of medieval fantasy and then add the creatures, places, and magic that make these worlds unique.

The worlds of the Dungeons & Dragons game exist within a vast cosmos called the multiverse, connected in strange and mysterious ways to one another and to other planes of existence, such as the Elemental Plane of Fire and the Infinite Depths of the Abyss. Within this multiverse are an endless variety of worlds. Many of them have been published as official settings for the D&D game. The legends of the Forgotten Realms, Dragonlance, Greyhawk, Dark Sun, Mystara, and Eberron settings are woven together in the fabric of the multiverse. Alongside these worlds are hundreds of thousands more, created by generations of D&D players for their own games. And amid all the richness of the multiverse, you might create a world of your own.

All these worlds share characteristics, but each world is set apart by its own history and cultures, distinctive monsters and races, fantastic geography, ancient dungeons, and scheming villains. Some races have unusual traits in different worlds. The halflings of the Dark Sun setting, for example, are jungle-dwelling cannibals, and the elves are desert nomads. Some worlds feature races unknown in other settings, such as Eberron’s warforged, soldiers created and imbued with life to fight in the Last War. Some worlds are dominated by one great story, like the War of the Lance that plays a central role in the Dragonlance setting. But they’re all D&D worlds, and you can use the rules here to create a character and play in any one of them.

Your DM might set the campaign on one of these worlds or on one that he or she created. Because there is so much diversity among the worlds of D&D, you should check with your DM about any house rules that will affect your play of the game. Ultimately, the Dungeon Master is the authority on the campaign and its setting, even if the setting is a published world.

Yeah thats what they wrote later.  I wouldn't particularly agree that they share the same Abyss such that Tiamat is the same Tiamat in each campaign setting.  I would say each campaign and each DM has their own Tiamat (if they want a Tiamat).

Just 1 Tiamat. But as a denizen of the outer planes she can manifest on an infinite number of prime material planes- what we humans like to call the many worlds.  The elemental inner planes, like the outer planes (unlike the prime material plane,) exists each as a single instance. Or at least that's been the story for the last 44 years.

Quote from Drawfonaf >>

Just 1 Tiamat. But as a denizen of the outer planes she can manifest on an infinite number of prime material planes- what we humans like to call the many worlds.  The elemental inner planes, like the outer planes (unlike the prime material plane,) exists each as a single instance. Or at least that's been the story for the last 44 years.

If that were the case then if she died in one campaign *on her home plane* then she dies for every DM and every campaign.

Campaign setting =/= campaign.

The idea they try to portray, at least in relevance to the OPs question about how different campaign settings work together, is that each setting acts like a universe amongst many - amongst a multiverse. Between these universes, these worlds, these planes.. Are transitional planes like the Ethereal Plane or the Astral Plane which can be used to travel between them. A party could, for example, use these to travel from Forgotten Realms to Eberron.

One world could also just be an alternate history version of another world, complete with its own versions of specific characters, like Tiamat. The DM is free to create this. Perhaps in one DM's world Tiamat was defeated. Perhaps in your world she wasn't. And perhaps it may be allowed for there to be a way for that first DM's party to travel into the second DM's world, purely to test themselves against Tiamat again.

Quote from Bogdogs >>

I'm somewhat confused on how all these different campaign settings fit together. Is each campaign setting entirely different, each with its own cosmologies as well, or are there multiple worlds on the material plane? 

You're confused because it is confusing. And ridiculous.

This goes back to the deep roots of AD&D, if not beyond... that to satisfy every possible DM, they made an omniverse in which every possible pantheon exists, and a putative prime material plane in which every D&D universe supposedly exists, even when some of these universes (like my current one) would specifically preclude the existence of most (or all) of the others.

I spent hours trying to make sense of the D&D omniverse to try and get it to be something coherent I could use as a cosmology for my world and I finally just gave up and did my own. I think the problem is they have had too many authors working on it from too many years, and on top of that for commercial purposes they need to basically say that all of these conflicting cosmologies are somehow simultaneously valid and extant, even though they could not possibly be.

So.. I basically just ignore it. Other than swiping an idea or two that I happen to like and that does not conflict with the cosmology I created. But that's the luxury of working with a homebrew world. If you are working with the published campaign settings, you basically have to live with the implicit contradictions that occur from claiming they somehow all exist in the same prime material plane while having separate and distinct cosmologies.

And thinking about the Multiverse... I'm surprised I've never used any of the Eternal Champions in any of my D&D Campaigns!!!
I mean, I flippin' freaked out the first time I saw Elric in a D&D tome.  But to pull any of the versions in (Corum or Hawkmoon most likely after Elric) as an NPC to complete an impossible thing...  as if I needed to create any more "characters" for my NPC blocks.. but... looks like this will happen... ha ha...

Every D&D world exists in the Prime Material Plane. It is up to the DM to decide which one(s) exist in the campaign they are running. It could be 1 or all or anything in between.  The idea of a multiverse makes things easier to reconcile when it come to lore, deities and the interactions with the planes, but there is no reason that they can't all exist in the same Prime Material Plane.   

The Material Plane

The Material Plane is the nexus where the philosophical and elemental forces that define the other planes collide in the jumbled existence of mortal life and mundane matter. All the worlds of D&D exist within the Material Plane, making it the starting point for most campaigns and adventures. The rest of the multiverse is defined in relation to the Material Plane.

This would imply that there is only one shared Material Plane.

There is one shared Material Plane, but each DM has their own version of it. But that means that if a DM runs campaigns in multiple worlds, all of those worlds exist in a shared universe.

In original D&D, it was assumed that all games ran in the same shared world, so players could take a character from one DM’s campaign to another’s no problemo. But as the fan base grew that became problematic so they invented the “Prime Material Plane” as a way to have every DM have their own world in a shared universe so that crossover was still possible, but there would be no canon issues.

Then they invented alternative cosmologies and everything went to heck in a hand basket. Now it’s best just not to think about it too much.

Quote from Kotath >>

 I have played every version of the game to date, including the original white box. I do not remember any such shared world.... 

 I have not, I started in the early ‘90s. But from what I heard from those who came before, it was like that. Here’s an example I happen to be able to reference:

Its hardly worth arguing about since everyone is using different terminology  to mean different things.

Do what you want. No one outside of your gaming group ultimately cares or is affected. That's not to sound harsh, but to say, don't sweat the small details. If you like something from one book, throw it into your world. If a player knows all details about such a thing, shift around some names for consistency.

Now, when folk outside your gaming group care (or are affected), like Critical Role or the words of Keith Baker (Eberron's creator), the "making stuff up" of D&D may be met with some push-back. It's like Game of Thrones. If George R.R. starts changing things around, all sorts of folk will ask, "is it canon?" What timeline?" "How does that affect my enjoyment of the franchise, etc."

At your table though. Shopping list books, worlds, settings, places, adventures, characters as you see fit until the story is filled with as much fun as possible.

All the worlds of D&D exist within the Material Plane

I know they mean this but that should read, "All the published worlds of D&D exist within the Material Plane." 

My world, for instance, does not. It exists in what I call the "Mortal Plane," which is defined in my setting as the only plane in which (normal) time and space exist. All the other planes in my omniverse have a strange, generally non-linear, version of time and space, or maybe none at all (depending on the plane). And in the Mortal Plane, there is only one universe, again, the mortal, "time and space" universe. The rest of it is all supernatural in some way.

The way I see it,  Eberron and FR can be on the same plane as they are different planets and the scholars of those worlds believe that the rules of magic on their world must apply to all worlds, but they are just wrong. Magic behaves differently on other worlds.

Why?

Because magic.  

Quote from Lyxen >>

Quote from Golaryn >>

The way I see it,  Eberron and FR can be on the same plane as they are different planets and the scholars of those worlds believe that the rules of magic on their world must apply to all worlds, but they are just wrong. Magic behaves differently on other worlds.

Why?

Because magic.  

 Huh, no. First, the solar systems of both worlds is actually fairly well documented and they are not on the same planet and even not in the same solar system. And it goes beyond that as both worlds have identified constellations that differ. After that, it's not a question of "the scholars believe". When a sourcebook explains to you how a given universe works, you are taking it for granted. The players and the scholars can have it wrong in the world, but the DM knows, and he applies what's in the sourcebook, assuming that he wants to be cannon.

 Golaryn never suggested they were the same planet. They wrote that both planets exist in the same Prime Material plane, but that they had different ideas about the cosmology of the planes.

They could be in completely different galaxies as far as what they wrote implies.

Quote from IamSposta >>

Quote from Lyxen >>

Quote from Golaryn >>

The way I see it,  Eberron and FR can be on the same plane as they are different planets and the scholars of those worlds believe that the rules of magic on their world must apply to all worlds, but they are just wrong. Magic behaves differently on other worlds.

Why?

Because magic.  

 Huh, no. First, the solar systems of both worlds is actually fairly well documented and they are not on the same planet and even not in the same solar system. And it goes beyond that as both worlds have identified constellations that differ. After that, it's not a question of "the scholars believe". When a sourcebook explains to you how a given universe works, you are taking it for granted. The players and the scholars can have it wrong in the world, but the DM knows, and he applies what's in the sourcebook, assuming that he wants to be cannon.

 Golaryn never suggested they were the same planet. They wrote that both planets exist in the same Prime Material plane, but that they had different ideas about the cosmology of the planes.

They could be in completely different galaxies as far as what they wrote implies.

 ^^this