The Normal Exit is, as the name implies, the exit a player will normally take. It is usually easy to find, and usually brings you through the game. The exception to this is in the Star World, where getting the Normal Exit of the levels sends Mario backwards.
There are three ways to complete a normal exit. One way is to hit a Goal Tape. Another way is collect a Goal Sphere. A third way is to defeat a boss.
There is one exception to the third way, though. In Donut Secret House, defeating the Big Boo Boss will unlock the Secret Exit instead of the Normal one.