Having a good idea of what your core anatomy consists of is a great way to help learn what you should be doing in terms of your workouts.
The first part of your core anatomy is your abdominal muscles. Most people know this as the 'six-pack' and while there are not six individual muscles in the abdominal area, there is a tendon that runs the length of it and from side to side that gives you the subdivided appearance. The muscle that makes this part of your core anatomy up is called the rectus abdominis and runs the entire length of the midsection.
To the sides of that, going in a diagonal direction are the oblique muscles, which help to support the rectus abdominis as well as perform twisting actions.
Lastly you have your transverse abdominis, which is an underlying muscle that aids in intra-abdominal pressure and pulls the abdominal wall inward.
On the other side of the body you have your erector spinae muscles, which are responsible for flexion, extension and rotation of the entire back. They run the length of the spinal chord and consist of many small muscles, all intertwined with each other.
Finally, making up the upper side portion of the body that will also assist the abdominal and back muscles in keeping you stable and performing a variety of actions are the latissiumus dorsi muscles. These attach right by the under arm and span across to your shoulder blade and then down the side of the body. They are the ones that will predominately work whenever you perform any type of pull-up or pull-down action.
So when training these muscles, really focus in on contracting and releasing the particular muscle group you are working at that moment in time. Doing so will ensure you see the greatest benefits and prevent you from using other muscles nearby to assist the action you are trying to accomplish.