How do they get caffeine out in decaffeinated coffee?
The most commonly used method is the Swiss Water Process:


A batch of unroasted beans is soaked in hot water.  Caffeine is water soluble, so it's extracted from the beans.  However, some of the other substances that give coffee its flavour ("coffee solids") are also water-soluble and are also extracted.
The first batch of beans is discarded and the caffeine is filtered out of the water/caffeine/coffee solids mixture by a carbon filter.
Another batch of unroasted beans is added to the hot water/coffee solids mixture.  The water is already saturated with coffee solids, so only the caffeine is extracted from this batch of beans.
The now-decaffeinated beans are removed, the water/caffeine/coffee solids mixture is filtered again to remove the caffeine and the process can be repeated with another batch of beans or the same batch to remove even more caffeine.


There's no universal definition of "Decaf," but it generally means beans that have had at least 97% of the caffeine (by weight) removed.