"Great achievement is usually born of great sacrifice, and is never the result of selfishness."
- – Napoleon Hill
"The amateurs discuss tactics; the professionals discuss logistics."
- – Napoleon Bonaparte
"And the madmen build a lighthouse in Hell!"
- – Also Napoleon Bonaparte
The Astronomican is a psychic marker pen the Emprah used to mark his patch of the galaxy, which is basically all of it.
The Astronomican is a device created by the God-Emperor of Mankind in preparation for the founding of the Imperium of Man. Following the Golden Age of Technology, humanity was cut off from its colony worlds due to violent Warp storms and the loss of its own beacon grid, rendering it unable to safely traverse the Warp. While the Warp storms died out around M29, it was all but impossible for Navigators to safely guide space ships through it. To solve this, the Emperor created the Astronomican, a device which acts as a psychic "lighthouse", allowing warp-travel in a 50,000 light year radius of Terra (you can still go outside that, but it will be difficult to navigate and you better hope the Gellar Field doesn't fizzle out). The Emperor eventually planned to phase out Warp travel completely with the construction of an Imperial Webway, but one guy accidentally ruined it, which led to an immeasurable amount of rage due to Not as planned, both from Big E. himself and the fan-base alike.
Following his entombment to the Golden Throne, the running of the Astronomican fell to the Adeptus Astronomica. It's overseen by the Master of the Astronomican, who has a permanent seat on the council with the High Lords of Terra. Given the sheer amount of psychic power contained within the device, the Emperor's mind is used to direct the power of the Astronomican. To keep it running, one thousand psykers are rounded up every day and sacrificed to power the Astronomican. Unlike the ones sacrificed to maintain the Golden Throne, these psykers are trained to keep it running and consider it a holy duty to sacrifice their lives... At least if you take things at face value like the Adeptus presents it. And while it is true their sacrifice is a vital necessity for the Imperium, a more cynical view is that the device is a trash compactor for unwanted psykers: those that are weak of mind, powerful but outright insane or just happened to piss someone off. By keeping it active, humanity has a guaranteed way of navigating the Warp but also an ostensibly laudable way of "depositing" dangerous psykers and keeping the Imperium "functioning".
Also, the "thousand a day" is something of a misnomer: there's actually way more than one thousand psykers inside the Astronomican Chamber of the Hollow Mountain (the former Mount Everest) at a time. The mechanisms of that chamber gradually drain each victim of all memories, thoughts, emotions, and eventually the soul itself through a course of a few weeks, or in case of the most unfortunate ones, months, and all this time for them is filled with extreme, incomprehensible, mind-blowing, "I wish I was abducted by the Dark Eldar" pain, until all that is left of their souls ceases to exist (so no afterlife for them) and all that is left of their bodies crumbles to dust. It is telling that even the Emperor refused to implement this solution when Malcador suggested it the first time and only reluctantly agreed to it when it became clear he could no longer both do everything he needed to do and power the Astronomican alone at the same time. Let that sink in for a minute: when the most ruthless warlord in the history of mankind, the guy with a lot of geno-/xenocides and war on galactic scale on his hands, when he thinks something is too extreme; you know you're in for a real treat on the Grimdark scale.
The Imperial Creed is a bit uncertain as to the exact theological nature of the Astronomican: the accepted line is that the Emperor maintains it by his sheer divinity, whilst the psykers are either purified of their sin and redeemed in death or are just not talked about. What few historians the Imperium has have learned not to point out that the Astronomican ran before the Emperor sat on the Golden Throne.
It lights up the warp, Council-of-Nikaea-beacon style, filling minds of psykers with holy singing, thoughts of martyrdom and images of Living Saints of course. While sacrificing the trillions upon trillions of humans on Terra. Typical dickery- though on second thought, there are many examples. The fact that the psykers are sacrificed and are not allowed to rest between sessions is also a typical source of grimdark.
The Talon of Horus book by Aaron Dembski-Bowden describes... interesting side effects of the Astronomican activity. Where its light collides with a potent Warp Storm, like the Eye of Terror or the Maelstrom, the mix of both creates the Firetide: a rim of space fucked up so hard even daemons are afraid of it. Not only do they get quickly burned by its walls of holy warp-fire, these are also inhabited by warp-spirits that could be described as flaming angels; just like Chaos daemons, they don't mind slaughtering, raping, and eating chaos worshipers (who, if they are very lucky, will suffer this fate in that order). These "angels" are known to cleanse their territory of all life oldcron-style, no matter its allegiance, but unlike daemons, these spirits cannot be reasoned with, as they are as insane from millennia of intolerable pain and suffering, probably just like the source that gave them life. The Firetide engulfs multiple "Radiant Worlds", planets touched by the golden psychic storm without being burned by it. In these regions, the Astronomican can manifest an avatar of the Emperor's will known as Imperious, an actually way nicer chap than these postal-going "angels" described earlier, taking the form of a humble pilgrim wearing a Scream mask of gold and light.
There's only one real drawback to having the Astronomican (you know, aside from whole sacrifice of a thousand innocents per day): it's attracting the Tyranids to Terra (they were already on their way to the Milky Way due to the Pharos) like a swarm of mosquitoes to an infra-red torch so it should really be renamed to the Astro-NOM-ican. Is this a drawback? Who knows? Maybe Big-E needs the 'nidz to set him free from his Golden prison. After all, their presence would likely close the warp rift in the Eternity Gate as it is so small and their Shadow in the Warp is so large. Additionally if you want an entire army to attack one point, you may as well have it attack the most heavily defended point you have. Holy Terra fits that bill so who knows how many planets are being spared because the nids are focusing on the Astronomican?
Another 'drawback' to the Astronomican is that it's a lighthouse, and every one can use it. In theory if the Tau or Q'Orl could develop the ability to 'see' the light the way Navigators can then they too could use it, though the effectiveness of the former might be limited due to their weak soul-presence, and the latter limited by their current existence in canon being dubious at best. There have been a case or two of chaos exploiting the light such as Shon'tu using a navigator, who of course would have had to have been using the Astronomican, to drop a hive ship within 100 light years (read: galactic spitting distances) of Terra.
That Other Thing
It's not mentioned much, but the Adeptus Astronomica is the single most important organization in the entire Imperium. Without it, there is no Imperium. Humanity is only able to leverage its numbers and make war on a galactic scale because of the reliability of its warp travel. Yes, sometimes ships are lost to the Warp or arrive two centuries too late. However, sometimes every successor chapter answers a call and arrives to bail out Baal or Terra all at once. Sometimes crusades of hundreds of ships go cleansing a path hundreds of star systems wide. No other race in the galaxy has the same combined numbers, reactiveness and coordination, and it is owed to the Astronomican. The last time Mankind was unable to make use of Warp, the first intergalactic empire Mankind had (this was before the Age of Strife, in fact it was the same government that ruled at the time of the Dark Age of Technology, implied to be some form of corrupt but functional and effective democracy) fell apart, entire planets lost contact with each other, Terra became a shitpit (And THAT forced the Emperor to come out of the shadows and begin his preparations to end the Age of Strife), AND it was even smaller than the Imperium of Man, AND that time period didn't have all the nasty shit that plagues the galaxy today.
Also warp travel being what it is sometimes it fucks up in a beneficial way and the fleet arrives several years before it left.
In all seriousness, the lore depicts response fleets arriving even while invaders are still off-loading troops or after the battle has peaked but before it can swing either way. This implies Imperial Warp travel is usually very reliable and extremely quick. The rapidity of the Great Crusade itself also supports this.
The Imperium Nihilo
New Black Library books by Guy Haley shed some more light (nerp derp puns) on the situation. The light of the Astonomican is able, sometimes, to slowly melt back the taint of lesser Warp Storms, and at one point Guilliman outright says that Warp travel in the post Great Rift era is harder in areas where the Astronomican isn't shining, even if there are no active Warp Storms going on. That implies that the light of the Astronomican itself is somehow, making the Warp less fucky to fly through, which might go some way to explain why the Dark Gods call Biggie E Anathema. He's been aggressively glowing at them for eleven thousand years or more. Remember, the Astronomican came online in M30.8**, and the book Plague Wars is set in M42.111, so that's eleven millennia of him peeing in their garden without stopping.