Walking between the mounts of Safa and Marwah is one of the integrals of Hajj. Originally, Safa and Marwah are two small mounts which are adjacent to the structure of the Haram Mosque, one at either end of the long hallway where one performs the Sa^y. The valley that existed between these two mounts was filled at one point in time. Hence, it is not there anymore.
The Sa^y must be performed after completing a tawaf (whether integral or sunnah). Ritual purity is not a condition for the validity of Sa^y. One must walk seven (7) times between Safa and Marwah. One starts at Safa, making sure that one is behind the mark indicating the original beginning of the mount and proceeds to Marwah, again making sure to reach the mark indicating the original beginning of the mount. This is considered one (1) round. Returning from Marwah to Safa is the second round. One may stop and rest during the walking, it is not a condition that one does not stop. The seventh (7th) round will end at Marwah.