A thousand years ago, there was a civilization on the Moon . . .
But that's not really relevant to the story. So let's skip forward a few centuries.
Once upon a time, a vampire came to England. He hypnotized a mental patient named Renfield, turned a fair young woman named Lucy, and tried to follow her up with Mina Harker (née Murray).
Not only did Mina resist the vampire's influence, she turned it against him. With the help of the learned Professor Abraham Van Helsing, as well as Lucy's distraught but capable trio of former suitors, the vampire was tracked down and overpowered. Once his influence over Mina was broken, Lucy's husband delivered the killing blow.
Or at least, that's what it says in the published version.
Thirteen-year-old Integra Hellsing loses her father, inherits the leadership of England's secret vampire-hunting organization, survives an attempted murder by her jealous uncle, and finds a vampire in the basement. All in one day.
In theory, Alucard is supposed to do whatever his Master orders. In practice, there's a reason he's been locked up since the end of World War II. He's a valuable weapon for the Hellsing Organization, but that doesn't mean he won't cause trouble. Like when, ten years later, he asked dying policewoman Seras Victoria if she wanted to be turned.
Now Integra has two vampires running around her mansion, and only Walter Dornez, family retainer and right-hand man, to help wrangle them.
Of course, when a rash of young, cheap, and reckless vampires starts plaguing the UK, Hellsing will need all the firepower it can get.
It doesn't help that they come into frequent conflict with the Iscariot Organization: the Catholic group dedicated to wiping out vampires, werewolves, monsters, Protestants, and other unholy creatures. Or that it turns out the dime-a-dozen vampires are being engineered by Millennium, a group of Nazi researchers that Walter and Alucard supposedly wiped out back in 1942. Or that, with all of Shine's crossovers, when it comes to other groups that might have it in for our heroes, the sky's the limit.
A Note On Hellsing Continuity
There are three separate continuities for the series: the 10-volume manga, the 13-episode TV series, and the ongoing OVA (Original Video Animation — in other words, a direct-to-video anime series).
When the TV series was made, only the beginning of the manga had been written. As such, it follows the plot of the first two volumes of the manga before making up its own ending, all while exploring original characters and subplots along the way. The OVA is much more faithful to the plot of the manga, but without the interesting side trips.
In the beginning, Shine was set in the TV series continuity. After a time travel storyline shook things up, it was relocated to a strange mishmash of the two. Characters that only appeared in the TV series often still exist; but if they died before the ending, that doesn't necessarily mean they're dead now. Don't worry too much about charting an exact timeline. Just roll with it as it comes.
Finally: There is no live-action Hellsing. There is not going to be a live-action Hellsing. Please stop asking.
The various other threats Hellsing faces are introduced as they come up in the comic. It's a big ol' multiverse out there.
From R.O.D: Read Or Die, we have I-jin, superpowered clones of historical figures. (For example, a clone of physicist Erwin Schrödinger with power over quantum probability.) These usually fall under the jurisdiction of British Library Special Ops, whose ace in the hole Yomiko Readman has the ability to control paper.
From Witch Hunter Robin, we have, well, witches. Paper users such as Yomiko are a particular variety of witch (called "craft users" when they're not hostile), but they can have pretty much any skill you might imagine. Global organization SOLOMON generally takes care of witchcraft-based threats.
From Revolutionary Girl Utena we have . . . whatever Anthy is. She's been labeled a witch since time immemorial, but somehow that label seems far, far too small.
From all around the world come legends of various types of vampires, demons, and other supernatural threats. Luckily, Hellsing has allies to match. From high-tech weapons dealer Miss Ann Warbucks (Little Orphan Annie) and colleagues Larry Vincent and Minnie May Hopkins (Gunsmith Cats) to Paris-based regenerator Madeline (Madeline de Paris) to mild-mannered bookstore owner Mr. A. Ziraphale (Good Omens), our heroes are by no means alone.
And Shine Heaven Now
In the early strips, Shine is almost entirely comedy-based. Rabid fangirls, desert islands, and Dr. Seuss homages are the fare of the day.
Over time the shorter stories give way to longer and more involved plots. The themes begin to darken and the foreshadowing picks up, as the various subplots weave their way towards the single epic battle that spanned the the last five volumes of Hellsing. Only in this version the stage will be global, and there will be an exponentially greater number of players, drawn from every series featured so far and then some.
Also, there will be regular punch lines. Because it's more fun that way.