how i understand it all
4th demension: 3D but there is a plane that connects 2 other planes together WITHOUT intersecting the other plane.
hyperspace (program tutorial but intro to 4d)
pictures to help u understand:
klein bottle
see how this is just one big tube thingy. in 4d the "handle" simply connects to the bottom, but since we can only display 3d images it passes thru the side connecting to the bottom. the 4th dimension connects the handle and the bottom with out intersecting the side.
more pictures and some movies