High time I reared my ugly head in here again with another post on worldbuilding!
This time around, let's talk fauna. Flora will be covered later (if desired and I feel like it. Lol) Not all settings require it, but even if you don't makin' up animals can be a lot of fun.

So, let's begin!
Probably an important part of this whole thing...
...Is environment. A creature with a lot of fur and thick hides would most likely not live in the most dry and desert-like of environments! Just like things like lizards or other similar animals wouldn't live in the cold. Mind you, you can make them this way if you so wish, just make sure they fit in their setting and you have an explanation. For example... Umm... Dragons. I know it isn't the greatest of 'em, but still.
They can survive in the cold due to the fact they can breathe fire. Volia, an explanation! Or... They hibernate like some frogs do, IE: freezing themselves and thawing themselves out in the spring.
So, an environment will directly effect a creature 90% of the time. This is very important to remember when you are creating these things! All creatures around the world are suited to survive in their environment. Some better then others, but the point remains valid. You want yours to be believable, you have to do the same thing.
Creature traits and stuff
The most important thing to remember here is that NOT ALL FICTIONAL CREATURES NEED TO HAVE SOMETHING ABOUT THEM THAT MAKES THEM FANTASTICAL! In other words, you could have small animals that could quite easily exist in our reality! However, this is primarily a fantasy forum, so let's keep it that way.
When I look to give creatures traits, I look to Darwin and the environment they live in. A creature would evolve to better survive, obviously. Using the dragon example from above, they evolved the ability to breathe fire because...? See what I mean? Though, at the same time, you could say the creatures were created by the gods (goddesses, god, the devil, some eldritch entity or even a bored writer like myself with too much free time on his hands!

)
Going off the created tangent... Well, better not. I'll cover that next.
BACK ON TOPIC: Creature traits from evolution. You're not gonna see a sloth without long claws to help it crawl up trees (though, I am not THAT knowledgeable on sloths, so there might be ones who don't have claws... ANYWAYS!) Just like you're not gonna see a human being without opposable thumbs. (Well, save for either amputation or a genetic defect.) So, think about the creature and how it lives in its environment. What would it need to survive?
Vestiges
Humans have tonsils from back whenever. They don't serve a purpose anymore, but they're there. Animals have these too, so why not think about throwing maybe one or two of these in there?

They served a purpose at one point in time, but no longer do.
Made in the lab
...Or by a divine being, whatever you want...
Creatures that have been 'engineered' this way have to follow one or two of the same rules: adapted to their environment. However, because you're more or less playing god when you make these kinds of animals, you can go ahead and just let your imagination run wild! Maybe through selective breeding and exposure to majiks, an animal has evolved a larger brain, and specific traits of its parents! Or maybe through extensive engineering and splicing, the creature is now capable of carrying a 'genetic memory' where everything the parent knows is passed onto the next generation!
So, that's about it. I am sure this isn't the only way, but it is a way I have used time and time again and found to be very effective.