1's complement addition is distinguished from the 2's complement addition typically encountered in (unsigned) computer arithmetic by how overflow bits are handled. 1's complement overflow bits are carried around back into the sum while 2's complement overflow bits are discarded.
In general, the inverse of a number under a given mathematical operation is the value which when operated on with that number returns the identity element. The 1's complement additive inverse of a number is its bitwise complement (replace 0s with 1s and 1s with 0s). This proposal relies on a number and its complement summing to zero (the additive identity element). Actually they sum to negative zero--1's complement addition has two identity elements. Recall that an identity element under a given operation is a value which leaves any other number unchanged when the operation is applied. Under 1's complement arithmetic the addition of either zero (all 0's) or negative zero (all 1's) to a number will generate a sum equal to the original number.
1's complement addition is both associative and commutative (it forms an Abelian group over the unsigned integers, ref. 11), so it is immaterial whether an identity element is added to a number or the number is added to an identity element, or whether the number operates on its inverse or the inverse operates on the number--both arrangements have the same result. Also note that the operation of subtraction is equivalent to adding the inverse (complement) of the number.