An improper rotation axis is defined by a rotation around an axis followed by a reflection across a plane perpendicular to that axis. The subscript in S6
means that you rotate by 360/6 = 60 degrees, and then apply the mirror. I'm not sure where the S comes from, but it is the symmetry symbol used for improper rotation symmetry. Also notice that there is no such thing as S2
symmetry because that is the same as i, which is the symbol for an inversion center.
As for point groups, they are symbols that define the overall symmetry of a molecule depending on the symmetry elements that are present (rotation axes, mirror planes, etc.). You usually don't study them until you get to inorganic chemistry or some higher level math classes. They have implications for some of the physical properties of compounds as well as bonding.
Don't confuse these point groups with crystallographic point groups or space groups, which are different.
Here is a flowchart for how to arrive at the various point groups: http://www.poshusta.chem.wsu.edu/GroupTheory/FlowChart.htm