Characters in games are, of course, extremely important. The character you play as is the conduit for which you experience the game itself. You play as that character, you tend to believe that you are them; you therefore take on their roles and motives throughout the game.
Game designers try to make the character you play believable and also relative to you by many different tactics. As you play, they make you feel what the characters feel. They try to get you to feel as confused as they are. Their emotions, by seeing a terrible tragedy happen, or experience some form of betrayal, are meant to be understood by you, the player. You can sympathise with their fear, anger, happiness, and as you get more attached to them, you start to become them. You want the character to succeed, and generally you want to help them to achieve a happy ending.