I just might have an unnatural fondness for alliteration. I'm also a nerd for posture. I think that proper posture is one of the underappreciated aspects of athleticism and of good martial arts technique.
Packing is a term used to refer to the appropriate alignment of a bodypart. People might use it to refer to other things, but I've only seen the term used in two places: the shoulders and the neck. To my knowledge, the term "packing the neck" is very recent, and I only heard of it a few days ago from Brett Contreras' blog (which is one of my current favorites). These terms come from the field of strength training, not martial arts, but I believe they are applicable.
What are they?
Packing the shoulder means drawing the shoulderblades backwards and together and down, towards the ground (if you're standing). Pavel Tsatsouline says you should feel as if you're sticking your shoulderblade into your back pocket. If you want a really nice feel for what this does to your shoulder, both in improving stability and power, do an overhead press with your shoulder packed, then try it with your shoulder hiked up near your ear. You should feel a big difference in strength.
Note that this is quite different from the shoulder position used by boxers, who lift their shoulders when punching (on purpose) to protect the jawline. I feel that a packed shoulder offers better transfer of power (from the hips), better handspeed, and better safety for the shoulder joint.
Packing the neck involves positioning the head properly. You tuck your chin slightly, as if looking downwards, but pull your head backwards, to the rear. Imagine you're trying to give yourself an exaggerated double chin. You tuck the chin but pull up on the top of the head, elongating the spine. For a pretty good idea of what I'm talking about look at pictures of Mas Oyama punching things, such as this one I pulled from this wonderful gallery:
You'll see a slightly similar position used in Muay Thai - chin tucked, head down, but they lower the head too much, hunching the shoulders and causing a power leak at the upper back. They do it for protection - keeping the chin tucked into your chest makes you harder to knock out, which is the same reasoning that boxers use for elevating the shoulder.
There are, broadly speaking, two reasons I think the shoulders and neck should be packed when practicing martial arts (I'm not necessarily suggesting you should walk around with your neck packed at all times - this isn't that kind of posture - but you should do so when sparring or practicing fighting techniques).
First, having a packed neck and shoulder put your body in the strongest, most injury- resistant position. A packed shoulder is more stable (better supported) and less likely to incur injury or dislocate. A packed neck puts your cervical spine in a good position for resisting forces, making neck injury less likely. It also literally puts your chin and jawline in a better protected position, possibly preventing a knockout. Think of what typically happens when someone kiais - how often does the chin lift and jutt out? Now think about doing that while someone is trying to punch you in the face. Not a good idea.
Second, a packed shoulder and neck are more powerful. You will transmit power better through packed joints than through loose joints. Don't believe me? This is a simple experiment - punch a heavy bag with your chin up and your shoulder hiked up, then with both packed. Or do some kind of pressing movement, with weight, and see how strong you are with and without packing. I think you'll see my point.
Another interesting experiment is to do your pushups, pullups, or any other upper body exertions, with your shoulders and neck packed and unpacked. I was shocked at how much easier packing my neck made my chinups and pullups.
To assist with packing, you might need to work on the strength of various muscle groups. If your lats are weak and your traps strong you might have trouble maintaining packed shoulders. If your neck extensors are weak you'll have trouble keeping your neck packed. Do pulldowns, chinups, or dip holds (get into a dip position and hold it with arms straight for a period of time) to get better at shoulder packing. For the neck I'll take a rubber band, loop it around my head, and press back against it for reps.
Take home: Try keeping your neck and shoulders "packed" whenever you practice techniques or perform strength movements. Make that posture automatic enough so that when you hit things during sparring you maintain a packed alignment. You'll be stronger and healthier!