A common objective in many games is to save the princess. She is the ultimate representative of your side, your society, your inability to self-govern. Without your beautiful and benevolent leader, your domain will spiral into chaos. Everywhere there will be careless turtles, fire breathing flowers, angry skulltulas.
Every hero and villain need a trophy to battle over. While an entire kingdom is unwieldy, it comes free with a single princess. Princesses seem to be quite easy to kidnap but difficult to rescue.
Most princesses are ungrateful and offer no more than a thank you before they run away back to the “security” of their castle once you’ve rescued them. But the knowledge that you’ve not only restored your monarch to her throne but also order to your homeland should be reward enough for you. You’re a hero, not a pop star.
Some awesome Princess-Savers that are worth checking out:
The first Legend of Zelda had Link trying to rescue Zelda from Ganon, and the second sees him trying to rescue another Zelda from a spell that causes her to sleep eternally. All the games after that, though they usually do have Zelda kidnapped at some point, do not make it the major driving force of the plot; those games which are an exception to this rule usually do not feature Zelda at all. (source)
Monsters have (Probably) Stolen Your Princess! Those fiendish and misshapen fools have (as far as we know) made off with your true love – and they haven’t even had the good grace to leave a shred of evidence to follow. Only sweeping accusations and devastating style can save her from the fate she will (possibly) face now!
Super Mario Brothers – Oh, the ever useless Princess Peach.
In Super Mario Bros., the player controls Mario (and in a two-player game, a second player acts as Mario’s brother Luigi) as he travels through the Mushroom Kingdom in order to rescue Princess Toadstool from the antagonist Bowser. (source)
In Fat Princess, the player joins one of two opposing groups consisting of up to 16 characters each, with the goal of rescuing their team’s princess from the other team’s castle. To make their opponents’ task more difficult, players can feed the captive princess cake slices. With each slice, she becomes heavier and harder to carry back to her respective castle. Over time, the effect of the cake wears off. (source)