Hard water is water with a high content of mineral ions (charged particles) in it, specifically calcium and magnesium. These are just present in the water supply in certain areas, particularly the southwest. These ions tend to stick to plumbing pipes and fixtures, as well as tile or glass in a bath or shower area, which results in a build-up of a hard, white scale over time. It also causes soaps and detergents to be less effective.
Water softening either removes these ions (in the case of a reverse osmosis system) or uses an ion exchanger to put salts into the water in place of these hard water deposits. This would either be sodium or potassium salt pellets that you would buy at a home improvement store to fill the water softener salt reservoir every so often. They also have water softener services that come by to replace your ion exchanger with a charged one every so often (this is like the Culligan soft water service).
The resulting water in an ion exchange system has salt in solution instead of the hard water ions. This makes the water feel "soft," and you need a lot less soap for taking a shower, shampooing, doing laundry, etc. But, if you use the sodium pellets, you now have a lot of sodium in your water supply and, if you use this for drinking or cooking, you can actually get too high a daily dose of sodium from your soft water (this can lead to high blood pressure, hypertension, and heart disease). That's why people use the potassium pellets instead.