Old Ones (Warhammer)
The 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). They put a warp-gate at each of the poles and shifted the planet's orbit to something nicer for their amphibian biology. All that's left of them in present day are the degenerate & decadent rulers of the Lizardmen armies, the Slann.
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, 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.
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.