well let's break this down into chunks (We're in my territory now... I know 3D modelling and such

)
How does animation work? Is it taken care of by the artist?
ok, that depends on what kind of animation. how it works is that (Using skeletal animation on this example) the 'bones' deform the mesh in the direction/rotation of the 'bone' that moves.
Yes. it is taken care of by the artist by first rigging the model (Making a 'skeleton' out of 'bones')
then the animator (often the same person that made the model) animates it using the 'bones'
ok. now that was a simple explanation. if you need/want more google is your friend. ^^*
Can I as the programmer use Irrlicht to animate a character?
it is technically possible. but it would most likley be a lot easier to animate it in a 3D modelling suite such as blender, Maya or 3DS MAX (Just some examples. there are more, a lot more.)
Can someone explain the animation process from scratch...
ok. I will explain it in the simplest way possible
Make model
Rig the model
move the bones around
?????
PROFIT!!
</endofsimpleexplanation>
First the 3D model is created...then how is it animated and what is the programmer's role in it? Also what kinds of files are used??
the programmer doesn't generally have a role when it comes to animating. though. if you are an indie game-dev. you will most likley be the graphics artist and the programmer.
the files that are used are listed in the documentation. I would advise you to read the irrlicht FAQ and all the orher official documentation.
Can Irrlicht only work with MD2/MD3 files? How do those files work? How can an animation be done with and also without MD files?
no, irrlicht can use a lot more than that!
how the files work. you'll have to specify what you mean with that.
it can be done using any 3D format that supports animation. b3d is an excellent example.