Young diagrams with n boxes characterize the conjugacy classes of Sn. 
=> irreps of Sn are also labeled by the Young diagrams.

length of row i = size of ith largest cycle in a conjugacy class of Sn.

A general young diagram is of the form:

#######
#####
##
#

The dimension of the diagram is: 
Fill the diagram out in the following way:

m (m+1) (m+2)....
(m-1) m ...
.
.
.

And multiply everything together.

divide by:

form the same young tableau.
In each box put the number of boxes to the right + the number down +1.
multiply them all together.

= dimension of corresponding irrep.