We're asked this very good question somewhat often here at Muggswigz. I personally love when I am asked this question because it gives me a chance to satisfy someone's curiosity about coffee and hopefully bring them to a greater appreciation of the beverage. I may also like it because it gives me a chance to brag. There are primarily six popular ways of brewing coffee (excluding espresso), each a permutation of the brewing variables - brewing temperature, introduction of the water to coffee, and separating the brewed liquor from the coffee grounds. These methods are Turkish brewing, concentrate brewing, percolating, vacuum brewing, drip brewing, and French Press brewing.
Middle Eastern, "Turkish" or "Greek" brewing involves boiling in water coffee that was ground into a very fine dust. Traditionally the coffee is often brewed (boiled) with large amounts of sugar, but it may be brewed without the sugar. Middle Easterners seem to like to add spice to their coffee, and their spice of choice is often cardamom. The coffee is not filtered from the liquor and one is left with a pungent, thick, and muddy brew. In the western world this method is more of an occasional indulgence as opposed to an everyday brew.
The next method, concentrate brewing, is very popular in Latin America and some other parts of the world, and is starting to make a commercial appearance in the US. In concentrate brewing, large amounts of coffee are brewed with little water to brew a concentrate, when one desires a cup of coffee, some of the concentrate is mixed with some hot water. The concentrate can either be brewed hot or cold. When brewing cold one must let the coffee sit for at least a day. This method results in a mild, light-bodied cup with little aroma, and often little acidity and a muted flavor.
Percolating, the procedure that involves continuous brewing of coffee grounds using boiling water which then turns to boiling coffee liquor brewing overextracted grounds. This method, while practical, is a disparaging disgrace to the coffee bean. Even brewing with boiling water is bad enough (coffee should be extracted at 195 - 205 degrees F), then actually boiling the liquor is asking for a thin, bitter, tarry cup. To add insult to a sufficient mangling, the grounds are continuously being overextracted. However, to show the variance of personal preference, I know of people that prefer this method. I can only imagine the preference can only stem from either positive memories associated with it, an acclimation to it over years of knowing no other, or the same phenomena that makes people stop to stare at a car wreck.