The S-flag is a latched version of sn-1. For example, in an 8-bit system, if the result of an addition is 101101002, then the S-flag is set. On the other hand, if the result of an addition is 001011102, then the S-flag is cleared.
The S-flag is the “sign” flag. Some instructions interpret this flag to see if the result of an addition (or other computation) is negative or not.
This is how an instruction can choose to interpret the value represented by a bit pattern. In this case, any instruction that reads the sign flag interprets bit patterns with the MSb (most-significant bit) being 1 as negative.
As an example, in a 4-bit system, a sum of 10012 results in the sign bit set. If an instruction (not add itself) interprets the sign flag, then 10012 = -C2(10012) = -01112 = -7.