Suppose we have a tensor
To contract any two indices is equate any two indices, thus summing over those indices.
If
is a tensor then we can contract
and
writing
If there are
indices in the supersript and
indices in the subscipt, with each index able to take
values then contracting one pair of indices will reduce the total order of the system from
to 
If
then we can contract the idices completely to give a sum ofterms.