The central and most important part of Shadow of the Colossus is not the main character, but the 16 Colossi that you engage in battle with. Each one has its own design and characteristics that sets it apart from the other fifteen colossi. They also have their own names, though to my knowledge these are never revealed in the game (Note: the names used here come from the Wikipedia article). The Colossi all have at least one weakspot, which is a glistening white design that can be seen only when the hero's sword is drawn. When struck on this spot, the Colossi takes a large amount of damage, versus the pitiful measly amount that a normal bowshot generates (which often does no damage at all).
Next we will look at how to defeat each Colossus. Note that for the Hard Mode battles, the battle is the same except for one or two new weakspots. This strategy will be discussed after the basics.