"Far, far below the deepest delving of the Dwarves, the world is gnawed by nameless things. Even Sauron knows them not. They are older than he. Now I have walked there, but I will bring no report to darken the light of day."
- Gandalf, Lord of the Rings
(WARNING: THIS ARTICLE MAY SCARE PEOPLE)
How to describe these grotesque mockeries of natural law? There are no words that can encompass such disgusting foulness, not in English or any other human tongue. They are The Other. The Inconceivable. Alien beyond comprehension, their sole existence is an affront to all reason. We could speak of painfully dissonant noises and nauseating colours, ichor-dripping vermiform tentacles and abyssal yonic voids, or complex mathematical geometries, but those are mere superficialities. Monstrous and sick though these stigmata are, they do not define the abominations; they are merely among some of the more common symptoms of their underlying wrongness.
Cosmic Abominations, are not simply creatures that look horrible on a metaphorically cosmic scale. They are alien madness-inducing reality warpers who literally ARE horrible on a cosmic scale. Eldritch is not just anything that looks like an ugly mashup of different kinds of Sea Animals. What actually defines the Cosmic Abominations is their defiance of natural law, as humans understand it. They are the things that should not be, the ultimate aliens. It is this what makes them abominable, that can use jedi style mind tricks on all but the strongest of those who encounter them. Such laws as they do obey, such as When The Planets Align or Sealed are sealed in an object, are their nature, not ours.
They are native to the genre known as Cosmic Horror Story, but they are not confined to it. Mild examples can be found throughout the horror and fantasy genres. Greater abominations can occur in almost any type of fiction. Usually they derive, at least in inspiration, from Lovecraft's work.
Physically, Cosmic Abominations range from almost human, through big ugly scary monsters, to the unimaginably bizarre. Generally, the weirder they look the more powerful they are, but this isn't a universal rule. Here, the "ugly" in "big, ugly scary monster" doesn't just mean that it's horrible to look at — it means that there's something about it, about the way it looks, or the spaces through which it moves, that violates every law of reality as you know it. "Big" doesn't just mean that it could use the Empire State Building as a toothpick — it means that the... thing doing just that is only the barest fraction of the monster's true form, the 3-dimensional tip of a multidimensional iceberg. Cosmic abominations are perhaps the biological equivalent of an Absolute Comparative: They are uglier, they are bigger, and they are more powerful than anything else in existence. Some abominations are so hideous that even lesser abominations find them abominable. Some look approximately human, but just reading their name condemns you to instant jedi style mind control. If you see one of these in the flesh, it's too late to run.
The most hideous of the abominations come from Beyond. Whether they are from beyond the stars, before the dawn of time, or a place incomprehensible to humans, they are alien to this universe and its laws. The creatures may actually outnumber humanity — trillions may dwell in abysses far below the ocean waves, trillions of trillions may drift between the stars — but they prefer wild and lonely places, where people seldom tread.
Slightly milder abominations can get an exemption, provided the reality-screwing does not override all of their rules and set a permanent stellar battle that anything that gets close dies horribly over a long period of time with so much excessive Mind control that repressing memories will be necessary. They are also typically the descendants of greater abominations, or the work of mad wizards or mad scientists who harnessed another dimension's powers. These include some of the rarer varieties of undead, so long as they are rare, and the product of ill-advised breeding programs.
Note that while normally used as an Antagonist, Cosmic Abominations can be helpful or neutral, though most of them are way beyond the whole good and evil system and are merely uncaring or incomprehensible, treating Earth as at best a colorful plaything, and at worst... they're absolutely insane planet destroying Maniacs.
If a Cosmic Abomination exists in a story where the primary antagonist is of a more human scale, it's probably the Bigger Bad. Even though their apperences may vary they usally look like an Octopus or a Squid.
By the way here's a Cosmic Abomination for cartoon characters: