A pitch class (pc) includes any pitch and any and all octave and enharmonic duplications. There are only 12 pcs--the notes of the chromatic scale. This invokes the concepts of octave and enharmonic equivalence in order to make all duplications the same pc.
For example: C5 is a pitch, while B#4, C2, and Dbb7 are all members of the same pitch class.
Pitch classes are often represented using integer notation. This is the practice of using the numbers 0 through 11 to represented the 12 possible pitch classes as follows: PC 0 = C, PC 1 = C#/Db, PC 2 = D, PC 3 = D#/Eb, etc.
A pitch class set is a collection of non-duplicated pcs. PC set have no fewer thatn three members, and rarely more than 9. {D,Eb,G} and {2,3,7} are two ways of representing the same pc set. Note the enclosing brackets and commas between members.
An interval class is a means of categorizing the distance between two pitch classes, still invoking the concepts of octave and enharmonic equivalence. There are six interval classes (ic). These include each interval from the m2 to the TT. Complementary intervals are considered members of the same interval class, therefore ic1 contains the m2 and the M7, ic2 contains the M2 and the m7, ic3 contains the m3 and the M6, etc.
A symmetrical set is one whose intervallic structure is the same whether you read it left to right or right to left. {0,1,3,4}, {2,3,5,7,8}, and {0,3,6,9} are all examples of symmetrical sets.