A magnet can be defined to have a north and south pole, in which magnetic fieldlines respectively leave or enter the surface of the magnet. But these magnetic fieldlines loop around and are self closing, which means that the lines entering the magnet connect with the lines leaving the magnet.
If you now were to cut the magnet in half, the magnetic fieldlines that ran through the magnet now are going out of or into the 2 parts of the magnet. These two ends now become the new north or south pole, depending on whether the already existing pole is south or north.
When you move the magnets away far enough, they fieldlines will now once again loop and close on themselves: you get 2 new magnets.
Hope this is clear enough, looking up some photos might help ;)
