Bleh, MGS4 was already mentioned.
With this criteria
Have you ever played a game that really made you feel drawn in to the story and really makes the emotions flow, a game that makes you really care about the story and characters, a game that you don't just play because it's fun or cool?"
The first game I can think of is that one.
Next in line is either Tales of the Abyss or Tales of Symphonia.