Generally this is done by loading animation data (built in a modelling/animation program like Maya) along with your model and reading the transformation and rotational data for each bone in a rig from there. This is commonly called skeletal animation.
Instead of loading the data from pre-baked animation files, you can adjust the bones procedurally (this would be referred to as procedural animation). Doing this well for very human-like behaviors such as walking, running, and jumping is an ongoing area of research.
There isn't a set of simple equations that you can apply to your bones to get even vaguely realistic motion out of them. Most procedural animation pipelines end up being an exercise in fiddling with a lot of little variables in order to achieve a desired result. You can probably find engines that support full or semi-procedural animation (the Fabric engine claims to), or procedural animation suites that integrate into modelling packages like Maya. There are also papers on the ACM and IEEE sites dealing with the subject, although they are behind membership paywalls so I can't link them directly.
The state of the art (as it is appropriate for and applicable to real-time simulations, like games) is not very great -- 100% procedural animation often looks a bit mechanical.
Now, this isn't to say that procedural animations are of no use; often they are combined with pre-cooked animations, for example by using inverse kinematics to allow for dynamic foot placement on irregular surfaces during idle animations (IK is essentially a method by which you specify the desired target point of the endpoint of a bone chain, and the appropriate displacement and rotation of the bones down the chain can be computed taking into account constraints and whatnot).