[quote=@BrokenPromise]
I've personally never seen a dire wolf called a monster. Not saying it doesn't happen.
My first reaction to seeing a question like this is to simply look at a dictionary. According to dictionary.com...

1. a legendary animal combining features of animal and human form or having the forms of various animals in combination, as a centaur, griffin, or sphinx.
2. any creature so ugly or monstrous as to frighten people.
3. any animal or human grotesquely deviating from the normal shape, behavior, or character.
4. a person who excites horror by wickedness, cruelty, etc.
5. any animal or thing huge in size.
6. Biology.
an animal or plant of abnormal form or structure, as from marked malformation or the absence of certain parts or organs.
a grossly anomalous fetus or infant, especially one that is not viable.
7. anything unnatural or monstrous.

As you can see, most if not all of Antarctic Termite's points were true. The word has been around for a long time, and has gained a lot of different meanings. The one over branching detail is that monsters have the power to scare most humans.
[/quote]

Yes, that is the dictionary definition. It does not answer my question at all and only helps very little. It kind of seems like you didn't actually read the question at all, only the title.
I'll go through them all to show you why they does not apply to this question.
Also, first place I looked. Most famous rpg-system in the world, a whole bunch of dire animals listed as monsters. Link: http://www.d20srd.org/indexes/monsters.htm

1. A legendary animal combining features of animal and human form or having the forms of various animals in combination.

I can see this as a subcategory of monsters. There are quite a few that seems to be some kind of mix between several other animals. However, would you really say that a sphinx or chimera is the same as a platypus? I would argue that they are different.

2. Any creature so ugly or monstrous as to frighten people.

Complete nonsense. I know lots of people who are afraid of spiders and snakes. Those two are the most common phobias in the world. Does that make spiders and snakes monsters in a fantasy setting? Because that is something I have never seen.

3. Any animal or human grotesquely deviating from the normal shape, behavior, or character.
And...
4. A person who excites horror by wickedness, cruelty, etc.
And...
6. Biology. An animal or plant of abnormal form or structure, as from marked malformation or the absence of certain parts or organs. 
A grossly anomalous fetus or infant, especially one that is not viable.

So a dragon is only a monster if it doesn't have wings? Because they are supposed to look like that. These definitions leaves every type of monster out except the mutants. 
Also, when people are referred to as monsters one generally talks about certain behaviour. My question is about the split between animal and monster, not society's view of correct or incorrect behaviour.
Only a small part of these definitions can be seen as relevant, the behaviour of an animal, which the gentleman below has covered in more depth.

5. Any animal or thing huge in size.

This point was addressed in the first post when I claimed that a dire wolf would be seen in the same way as we see a grizzly bear. 

7. Anything unnatural or monstrous.

Have to love the “thing - something like a thing” definition. This one doesn’t work either both because the definition is restating itself, but also because of what I said in the op:
[quote=]
Animals are the creatures that exist in the real world and monsters are the ones that doesn't. But how about the characters in the game/book/movie? Surely, to them the difference between a wolf and a dire wolf is just size, just like between a bear and a grizzly bear. So how does one define a monster in a world where both animals and monsters are natural?
[/quote]


[quote=@Gentlemanvaultboy]
Here's what I came up with after thinking on it for a while:
A monster is any non-sentient creature that goes out of its way to menace and attack other creatures in and outside of its ecosystem for reasons that are not natural. Signs that you may be dealing with a monster are:

1. An inordinate number of the creatures kills are not consumed or used in any way.
2. The creature is actively destructive to its ecosystem and/or environment.
3. The creature demonstrates a willingness to go above and beyond what would be expected of a predator in pursuit of food, such as a willingness to scale fortified walls to kill those living within or attacking armed sentients when it is clearly strong enough to hunt for easier prey.
4. Persistence to the point of recklessness.
5. A willingness to incur injury in order to attack others.
6. A willingness to fight on even having sustained wounds that would cause a normal animal to abandon its hunt.
7. Any behavior that can be construed as "cruel" or "malicious" on the part of the creature where a rational explanation for the behavior can not be found.
[/quote]
 
I really like this list you've made, there has been some thought put into this one. Just to boil it down, you're saying that the line between animal and monster lies not in its features or heritage, but behaviour. It’s a very interesting thought. The only one I could go against is really number two, could you elaborate on this? Also, would you say that monsters cannot be reasoned with?