Ohhhh boy, here we go:
So, in the beginning, the Warp was more removed from the material universe, and it was calm. Souls still penetrated into the Warp but there wasn't as much turbulence from emotions as there is now. Creatures still lived in there, and they were Warp Entities
Warp Entities include both things made of Warpmatter, like Vampires and physical creatures that spend the majority of their lives in the Warp, like Enslavers
Then the War in Heaven came, unleashing massive destructive weapons AND birthing the Eldar and the Krorks. The weapons unleashed during the War in heaven churned up the Warp and made it dangerous and volatile - it brought
Chaos
This riled up the Enslavers and god knows what else and that's how the Old Ones died, because an incredibly psychic race pissing off creatures that can turn psykers into Warp Portals is a bad idea
With the influx of new souls and the turbulence of the warp there started to be currents and eddies in the Warp formed by different emotions - Warpmatter is heavily influenced by thoughts and emotions, and like attracts like, so eventually these streams of similarly-aligned Warpmatter flowed together
The thing about warpmatter is that in large enough amounts it displays intelligence, and in very large amounts it displays Sapience.
Another thing about the Warp is as well as being affected by the emotions of the material races, it affects them in turn. So a large enough concentration of warp matter can become a self sustaining maelstrom of emotion -
a Chaos God
Most of these gods dissipate over time, in fact an infinite number of them are being created and destroyed constantly as they fail to gather enough energy and influence to become self-sustaining, but the three (later 4) largest ones are referred to as the Chaos Powers. Because they are self-sustaining, the Chaos Powers can split off a portion of their warp essence into semi-independent creatures with their own intelligence (Daemons) or imbue a Mortal with enough of their essence that they essentially become a warp entity themselves (Daemon Princes). Lesser Gods and even independent daemons can do this too, but it takes up a lot of their essence to the poiunt of being dangerous
So:
Warp Entity: Anything that spends the majority of its life in the Warp, even if it's not made of warpmatter
Chaos God: A maelstrom of warp energy formed by the collection of a large amount of aligned warpmatter into a maelstrom of energy to the point where it develops sentience
Chaos Power: A Chaos God that is powerful enough to become self-sustaining
Daemon: Fragment of a God given intelligence and semi-independent action, or an independent collection of warpmatter formed with its own purpose (Also a Warp Entity)
Daemon Prince: Mortal soul imbued with Warp Energy to the point where it can function independently from its original physical body
Now, that's pretty simple for the ones we think of as Chaos Gods, but what about other races?
Gork and Mork: They
are Chaos Gods, in that they are formed in the same way as the others. This was explicitly stated in Waaargh! The Orks. They share some aspects with the other Gods and if Orks stray too much out of the Orky domain they may end up feeding Khorne.
The Eldar Gods: Never explicitly stated. They do some decidedly un-warpish things like directly interacting with their followers and creating literal physical artefacts, and being shattered into physical remnants, but at the same time they also interact with the Chaos Powers on a somewhat even level.
Liber Chaotica has this extract that seems to imply they are Warp Gods created by the Eldar, but bear in mind this is from
an insane man from another universe relaying visions
I watched as the First Ones encouraged the younger race to reach further into the other realm, and with their vibrant minds and passionate souls create beings of power to fight the star gods. But the battle was long and the First Ones were now few, and as their numbers dwindled so too did their influence over their young creations. Without the wisdom and might of the First Ones to bind them, I saw the Eldar's warp-beings evolve from sentient weapons into living gods – the first true gods of the Immaterium. How I wept when the Eldar embraced them as such.
Personally I think they're more like Warp Entities that had the ability to take physical form. We know the Old Ones had Warp-based technology, I don't think it's inconceivable that the Eldar Gods had physical war-forms that could house them, using technology similar to Wraith-constructs. They would need to have a physical presence because they also fought with...
The C'tan: These are creatures that feed off of stars. A completely distinct category of creature from the others, they do not interact with the Warp at all, in fact it is anathema to them. I barely even classify these as Gods really, but they were given that role by the Necrontyr and they have control of the fundamental physics of the Material universe in a similar way to the Chaos Gods' control over the Warp, so I guess by that definition they'd count.
The Emperor: Bunch of Shaman souls in a flesh container. He's effectively a nascent Chaos God in the process of being born, but still tied to a physical body. He's accumulating his "belief" and able to manifest in some ways as miracles or living saints, but he's not fully formed yet. The Inquisition has and is fighting wars over what will happen when he finally snuffs it, whether that's a good thing for humanity and
whether they should kill him themselves to kickstart the process. It's not called "The battle for the Emperor's Soul" for nothing
The Hive Mind: The collected gestalt psychic presence of an innumerable number of individual beings, acting with common purpose. Appears as a "shadow in the Warp" that can disrupt communication and interfere with daemons manifesting (sometimes, varies by author). DID NOT LIKE the Great Rift opening up, experienced something akin to pain. Not a Chaos God by technical definition because it's not a maelstrom of energy, it's more like a roaming cloud.