Old Ones (Warhammer)
Old Ones of Warhammer Fantasy Battle and Warhammer 40,000 were major actors in the deep backstory of both worlds. There is some debate about whether they are actually the same people in each.
Warhammer Fantasy
In Warhammer Fantasy Battle, the Old Ones are psychic frog-men from beyond the stars (big silver space-ships are mentioned even in the latest main rulebooks). Knowing Chaos would one day appear in this reality and attempt to consume it (how they knew is unspecified) they decided to create a race of beings capable of keeping Chaos at bay forever, or even beat them (we don't know how ambitious they were). They reshaped the world, during which they created the Lizardmen races to aid them. The Slann were simply less intelligent reproductions of themselves, the Saurus were their bodyguards and had very few thoughts in their head other than whatever their current task is and the general calm reverence of the Slann and Old Ones (so born Space Marines), and the Skinks as the tiny frantic servants to the Slann. Requiring more magical power, they established Warp Gates at the north and south poles of the world, through which they were able to travel freely between the world and wherever they came from. They also siphoned pure (AKA formless raw potential, pre-Chaos God alignment) Chaos magic from them, which made their attempt to prevent Chaos from coming into a self-fulfilling prophesy. They made peace with the local natural intelligent life, the dragons, and educated them as best they could. This knowledge would latter pass onto the first created race of the Old Ones as the two races would become BFFs. Old Ones encounter greenskins around this time, although it's unknown if they were a creation or a pre-existing pest. They tasked the Lizardmen with wiping them out. Guess how well that went?
First, they created Elves. They shaped a continent called Ulthuan which actually floats above the ocean like a giant ship for them to dwell on, and instructed the Slann to teach the earliest of them magic (it's unknown if the Elven gods were formed by the Slann or Old Ones, or if they came later out of belief; evidence supports the former as Isha is the mother of all mortal elves with Kurnous and as such the gods may have been proto-elves). As they surveyed the Elven race, they decided the Elves could not beat Chaos alone due to being frail and having a low birth rate despite potential for mastery of magic and a VERY strong resistance to the Warp.
Next, Dwarfs were created. The Dwarfs actually resisted magic innately and in their presence it shifts back into formless current and seeps back into the Warp. The Dwarfs were hardy, and could physically bully magic into doing what they wanted it to (hence becoming great smiths of magic weapons that are free from Chaos taint using runes). However, the Dwarfs were stubborn and would not adapt fast enough in the face of demon trickery. The Old Ones abandoned yet another of their creations.
The next race created was humanity (in the heart of Africa), which combined the magical aptitude of the elves (to a lesser degree) with the durability of the dwarves (to a lesser degree). Humanity's short lifespan with rapid reproduction along with ambition and lust not seen in their elder races combined would enable humans to have time to adapt and overcome anything Chaos could send at them. But humanity was as weak-willed as it was strong, and was VERY weak to Chaos taint both in mind and body.
The Old Ones created Halflings next, although nobody really knows why. They were resistant to mutation but thats about it. Small, frail, cowardly and unassuming, they were an abysmal failure as Halflings aren't really good at much other than cooking and eating. In a way they were the proto-Ogres.
Various other races were created, more or less filling the gap of both anything added to the game down the road as well as anything fans can come up with. Likely that many of these pro to-races were wiped out by the lizardmen, with only the most "stable" being allowed to exist (men, elves, dwarfs)
Finally, the Ogres were created by combining the best aspects of Men, Dwarfs and halflings. They were highly resistant to Chaos yet still possessing magical talent, having a short enough lifespan with a reproduction rate they'd be able to outlast Chaos, being highly durable, and finally stubborn enough to not fall to Chaos willingly in most cases. In addition to this, they are able to survive on just about any food, from meat to tree trunks to rocks. As the Old Ones prepared to endow their newest masterpiece with gifts, the Warp Gates collapsed and Chaos entered the world. The Ogres, left with a growing population in a relatively stable environment, were not gifted with any intelligence or taught culture. As a result, the race most capable of defeating Chaos permanently as a unified force became tribes of increasing size which spread out across the world and served every faction (including Chaos and it's servants). The minions of the Four (who existed as they were BEFORE they were connected to the world, as Fantasy Chaos has no Genesis) spread out across the globe and wreaked havoc.
It's unknown what happened to the Old Ones. It's possible that they fell into the Warp and were consumed. It's possible they fled the world, leaving it to it's fate as they tried again elsewhere. It's also possible they remain in hiding, either guiding the mortal races to grow in strength and thwart Chaos or watching as their master plan unfolds leading to true victory. Either way, they're gone.
Lizardmen began worshiping the Old Ones as gods, with Slann pondering the messages left by their masters on golden tablets (which is apparently how all Old One communications were relayed) to the smallest detail, and acted on interpretation. At times this has resulted in counter-productive things like an attempt at world recreation which caused most of the Dwarf race to die and greenskins take over their fortresses and breed in numbers never before seen. At other times great things have happened according to plan, such as when they have prepared for and beheld the coming of a god prophesied by the Old Ones, who immediately consumed the minions of a rogue Greater Daemon/minor Chaos God and drove him from the material plane.
At some point, one of the resident non-Chaos villains learned the words the Old Ones used to shape the world. With it (at GREAT physical and spiritual taxation of himself) he reduced fortresses the size of modern cities into atoms.
In older canon, the Old Ones did not create or predate the other races (in fact, one old lich mentions remembering the coming of the Old Ones with disdain as something that came long after his first death). Instead they were simply space travelers, and the Slann were their retarded and decadent inbred descendants.
Warhammer 40,000
In Warhammer 40,000, the Old Ones were a race or group of races that were among the very first intelligent races in the galaxy, if not the first. Usually, the Old Ones are referred to as a single, specific race. However, Codex: Eldar 4th Edition, as well as some of the Deathwatch supplements, claim that they were actually a group of the earliest races, who all worked together.
They were incredibly potent psykers (especially since the Warp was calm back in those days), had access to the Webway (and may have built or discovered and expanded it), and had supremely advanced genetic engineering techniques. They created the Eldar, Krork (likely the ancestors of modern Orks), the earliest ape ancestors of humans, Jokaero, the Hrud, the Rashan, the K'nib, and many others; some of these were created solely as warrior-races, whereas many (like the human predecessors) were created merely as part of planetary ecosystems, or maybe just for shits and giggles. They were also immortal, a gift they seem to have given in some form to some of their creations.
However, it's basically impossible to talk about them without talking about the Necrontyr. The Necrontyr had grown up beneath a wretched sun that poured radiation upon them (this radiation may in fact have been Star Vampire shit, more on them later), so their lives were, without exception, incredibly short (although it's not clear how short). However, they managed to scrape together a small interstellar empire, with different star systems ruled by different dynasties. However, with the First War of Secession, the empire fell into infighting, as the dynasties violently jockeyed for power.
The Triarch decided a war was just what the doctor ordered to keep the Necrontyr together, and so declared war on the Old Ones, claiming the casus belli of "you didn't share immortality with us poor, pitiful, mortal creatures." This was, as would be expected, a popular cause, and started the so-called War in Heaven.
However, despite having much greater numbers and mastery of technology which matched or even exceeded that held by the Old Ones, the Necrons did not have access to the Webway, and so were constantly outmaneuvered. Soon, the Necrontyr were scattered and left with only a few coreworlds to the galactic north. With the War in Heaven collapsing, the Second War of Secession began, and the Necrontyr began to tear themselves apart again.
Before Necrontyr society collapsed entirely, however, the Necrontyr discovered the C'tan, the Star Gods, leaching on some of their stars. The Necrontyr made metallic bodies for the C'tan, which barely contained their incredible power, as they were basically material universe's equivalent to the Chaos Gods.
The C'tan known as the Messenger (later the Deceiver) offered the Necrontyr a solution: living-metal bodies that were nearly indestructible, in exchange for their service to the C'tan. The Necrontyr accepted, only for them to realize too late that their new bodies required the destruction of their souls and their enslavement to the C'tan. For some reason, the Necrontyr were also renamed the Necrons at this point. One of the C'tan, known as the "Burning One," also helped the newly-robotic Necrons enter the Webway using the Dolmen Gates. With the field leveled, the Necrons were now able to take the fight to the Old Ones.
At this point, the Old Ones created many of the aforementioned warrior races, especially the Eldar and the Krork, but even these couldn't stop the Necrons. The wild misuse of the Warp had also spawned the Enslavers (or, at least, let them enter the material universe), obliterating many of the Old Ones. (The birth of Chaos caused by the corruption of the Warp by the dying emanations of countless warrior races probably didn't help either, to say the least. Incidentally, this also means that the current state of Grimdark in the galaxy is all their fault.) With the Old Ones thus weakened, the Necrons and their C'tan allies were able to annihilate the rest of thems, at least according to Codex: Necrons 5th Edition. After that, the Necrons exploited the weakened state of the C'tan and turned on their masters; while they were successful in breaking them into countless "shards" of their former power and permanently killed at least one, they feared the retaliation of the Eldar and the other surviving warrior races and went into hibernation until they were ready to rebuild their empire.
However, although most races aren't aware of their Old Ones heritage, they still made a great impact on the nascent races. In particular, at least according to Xenology, many races, including the Eldar, have very similar gods, including a bloody-handed warrior, a great smith, a life-giving goddess of the harvest, a laughing god, etc. It seems that these gods were inspired by the Old Ones, although whether they're based on particular Old Ones or just general concepts of what they did is uncertain. If you read the examples of gods above and thought by yourself: "Haven't I heard of this in real ancient religions at some point?", and if you can then connect that revelation to The Emperor, congratulations - You are paying attention.
Also, in Hrud mythology (again, according to Xenology), there's one last Old One named Qah that only left 500,000 years ago (i.e., long after the Necrons entered hibernation), promising to return when the Hrud reunite to fight the C'tan again. This would imply that, in fact, the Old Ones were not destroyed entirely by the Necrons, and a few hid or escaped. If Fantasy takes place around the same time as 40k (as is occasionally implied -- see below), this would mean the Old Ones were active only 7,000 - 10,000 years ago, which also implies that they were not all destroyed by the War in Heaven.
Connections
In very early editions of Warhammer Fantasy Battle and Warhammer 40,000 (i.e. when the former was still called "Warhammer" because it was the only Warhammer, and when the latter was still called Rogue Trader), it was implied that the two universes were very closely linked, if not one and the same. Chaos warbands in Fantasy could get their hands on bolters and powered armor, and Beastmen and Space Dwarfs (called Squats) were regular parts of Imperial and Chaos armies in 40K. The Old Ones were described as coming from space even in the 8th edition Fantasy rulebook, and the 3rd edition Necron codex suggested using Lizardmen minifigures to represent their descendents.