From Donald Knuth's "The Art of Computer Programming" (Volume 2, 3rd edition, pages 203-204):
A two's complement number is complemented with respect to a single power of 2, while a ones' complement number is complemented with respect to a long sequence of 1s.
As an example, we take the four-bit number 0110, decimal 6, and calculate the binary representation of its negative value, -6. We'll use both methods: two's and ones' complement.
Two's complement
We use "a single power of 2", namely 2⁴ (1 0000), to get the two's complement of our example number 0110:
1 0000 - 0110 -------- 1010 The "two" in "two's complement" refers to the base of the number (2⁴) from which we subtract our positive number.
See also Wikipedia's Two's complement.
Ones' complement
The "long sequence of 1s" is four ones in our example since our input number 0110 has four bits:
1111 - 0110 -------- 1001 The "ones" in "ones' complement" refers to the digits of the number (1111) from which we subtract our positive number.
I've created this page if you want to play around with different ways of representing negative numbers in binary.