New Game Plus
After "beating" Dark Souls, more specifically the final boss Gwyn, the credits will roll and you will be brought back to the beginning of a new game. This is known as NG+ (or New Game Plus), with the + at the end increasing with each playthrough.
Souls gained by killing bosses in NG+ are multiplied any where from 2x to 5x what they gave you in your first play through.
You can play through the game a total of eight times, including the first play through. This means you can eventually go to NG + 7, Or NG+++++++. After this, the game stops scaling, though you can keep beating it infinitely.
You get to keep the Soul Level you were at when you Defeated the Lord of Cinders, and all the stats you worked so hard to raise. So starting over isn't nearly a rough as starting a new game. However, certain things do not roll over, All keys and specialty items such as repair boxes have to be repurchased or picked up again. The Rite of Kindling has to earned again, but any bonfires you have kindled stay at the level they were kindled to.