Unlike the favorite answer, I would venture and say that this is actually quite a big task if you refer to having a character that have animaton to it.
The end result would be to enable most of functionality tied via script and config files (in other words - data driven), but the process to build such a system is not simple.
Usually you start with an external modeler where you do the following:
1. construct a skin model.
2. construct a skeleton and attach to the skin according to weights etc..
3. apply all textures and material as desired.
4. export to your engine.
Now you can start build your animation bank for this character skeleton or any similar one if you associate it to more than one (usually that's the case if the skeleton is close enough or modular).
Some of the animations will contain only partial hierarchy (to be able to blend only part of the animation - for example, lower body is running while the upper body can use a different animation or inverse kinematics - shooting at an enemy during the run is a good example for that).
Now finally you get to the code part.
You need to build a system that will associate the animations to the skeleton and render it correctly.
This system also needs to be able to blend between animations and IK (for example - shifting between standing to running), and be able to combine several animation and partial animations - for example, walking (full body animation), looking at someone (IK for the upper body), waving a hand (IK with partial animation) and breathing (partial animation) at the same time would combine 4 different partial or full blendings.
I did not mention the fact that you need to build the importer and there is also a difference between action and the animation for the action. Action can have several animations combined or picked randomly - they will be associated via a data mapping file.
I would suggest taking a look at XNA - it has many examples, importers and much more and contains a lot of the basic stuff in libraries so you don't have to start from scratch.
As people also mentioned - Blender3D is a good choice for editor+ and so might be Torque just for the experience of a game engine.
One last word - if you are going to try it out, it's not going to be a walk in the park, but hey - so much fun and the results will make you proud of yourself.