Remove a few outer leaves of the head of cabbage and reserve.
Cut the head of cabbage in half and cut out the core. Reserve the core.
Slice each half of the head of cabbage into thin strips.
Place the shredded cabbage and the salt in a bowl and pound with a "kraut pounder" (see video) or similar utensil, such as a potato masher, for approximately 5 minutes until the cabbage begins to release some of its juices.
Place the shredded cabbage and salt mixture in a half gallon sized jar that has a lid.
Quarter the apple and remove the core and seeds. Coarsely chop and add to a blender.
Coarsely chop the core of the cabbage and also add to the blender.
Add water to the blender sufficient to cover the chopped apple and cabbage core. Whirl in the blender to make a slurry.
Add the apple/cabbage core slurry to the shredded cabbage and salt mixture in the half gallon jar.
Stir the mixture together in the jar and then press it down to compact it. (If you prefer, you can do this in a bowl and then transfer it to the jar. However, I do find it is better to do this in the jar so that you are assured of having the correct amount of cabbage with the correct amount of salt. If you do this in a bowl, you may not be able to precisely know how much cabbage will fit into the jar.)
Take the reserved cabbage leaves and fold them and put them into the half gallon jar on top of the shredded cabbage mixture.
Take a small 4-ounce glass jelly jar and put it into the half gallon jar on top of the folded cabbage leaves. The jelly jar will work as a weight to hold the entire mixture underwater.
Add additional water, if needed, to reach the neck of the half gallon jar allowing for approximately 1 inch of headspace. Place the lid on the jar.
Place the filled half gallon jar in an undisturbed place such as the corner of a kitchen counter, on top of a refrigerator, or in a cabinet or pantry that has a room temperature range somewhere between 68°F and 72°F. (SEE "RECIPE NOTES" BELOW.) The jar should also be out of direct sunlight as ferments do not like temperature fluctuations.
After a few days, the cabbage should begin to ferment and you should see bubbles in the jar. Release the cap of the jar to allow some of the carbon dioxide, produced by the fermentation process, to be released. Re-tighten the jar lid.
Re-check the jar every day and release some of the carbon dioxide by loosening the lid. Then re-tighten the lid.
After 7 days, taste the Sauerkraut. Keep in mind that it will continue to ferment once placed in the refrigerator. If you like the taste, refrigerate it. It's now ready to enjoy. If you are not satisfied with the level of fermentation, allow it to continue to ferment up to 14 days at room temperature before refrigerating. I do not recommend fermenting the cabbage longer than 14 days as it may become quite soft and less palatable.
Please note that initially vegetable ferments can taste a bit salty. However, over time, the ferment will become less salty as the vegetable - in this case cabbage - absorbs the brine creating a more flavorful vegetable, and the brine clinging to the ferment will taste less salty.
As to storage, vegetable ferments need to be stored at 40°F. This can be on the top shelf of your refrigerator or in the door of your refrigerator. They can not be stored at room temperature.