The one's complement of a binary number is what you get by flipping every bit: each 0 becomes 1 and each 1 becomes 0. 0101 → 1010.
It matters in two ways.
As a step. One's complement is half of twos complement negation. For any n-bit x, inverting gives (2ⁿ − 1) − x, which as a two's complement value is −x − 1. Add 1 and you have −x. In an adder subtractor, the XOR controlled inverters on B produce the one's complement, and the carry-in C0 = 1 adds the missing 1. Forget the carry-in and every difference comes out one too small.
As a format. Some early computers stored negative numbers this way, so −5 in 4 bits was 1010. Like sign magnitude, it has two zeros (0000 and 1111), and addition needs an extra correction step. Modern integer hardware uses twos complement instead.
The name means "complement with respect to all 1s": subtracting x from 11…1 never needs a borrow, and the result is every bit of x flipped.