Hand rails are the long part along the stairs that you hold when walking up or down.
Balusters
Balusters are vertical posts that rest in the base and the railing. Though, these do not provide substantial structural support they do prevent people and objects from falling through.
Base/Shoerail
The long section at the bottom that that parallels the railings.
Newel
Newels are vertical support posts at the end of the staircase railing, these are both functional and decorative.