The Barrier is a strong magical force separating the Underground from the Surface. The humans put this in place after The War to keep the monsters trapped within the Underground and the rest of humanity safe on the Surface.
Any human can fall through the barrier on Mount Ebott into the Underground, leaving them in the Ruins, where Toriel generally finds them. However, only a being with the SOULs of a human and a monster can pass through from the Underground out to the Surface. The barrier can be broken, but seven human SOULs are required to do so. Only eight humans have ever fallen through the barrier: the first human, the six humans whose SOULs were collected by Asgore, and the protagonist.