Endless staircase is one of basic impossible figures which is also known as impossible staircase. If you would raise up by the stairs then gone four flights of stairs he after you had turned to the staring point. You can coninue raising infinitely but have not moved from staring point.
The model of endless staircase was thinked out by english biologist Lionel Penrose with his son mathematician Roger Penrose. Endless starcase was the first impossible figure which was used by holland artist M.C. Escher in his lithograph "Ascending and Descending". A misterious monastery with roof in the form of endless staircase was painted on the lithograph. Every day monks are raising and descending by the staircase doing daily ritual.
Although Lionel and Roger Penrose gave renown to the endless staircase, it was discovered several years before by swedish artist Oscar Reutersvärd.
Let examine endless staircase at the image below. We start from the utmost left footstep colored red. Going clockwise we raise by the stairs and going anticlockwise we descend. Raising we leave red level below. But gone four flights we return to the starting point and see that there's two red levels at this point. So, we are at the red level and higher red level at the same time. This is the paradox of endless staircase.
You can think that the endless staircase cannot be constructed in reality. This is not the case. As all other impossible figures endless staircase can be constructed in the real world. But this construction will have gaps and it will look like impossible from single point of view only. You can read more about realization of impossible figures in the article "Impossible figures in the real world".