In two's complement the highest bit has a negative weight, −2, while all other bits have their usual positive weights. That is why the highest bit is also the sign bit: 0 for positive numbers and zero, 1 for negative numbers.
Sign bit. Every negative number starts with 1, and every positive number and zero start with 0.
Only one zero. Zero has a single pattern (0000), so no pattern is wasted.
Range. With n bits we store numbers from −2n−1 to 2n−1 − 1. That is −8 to 7 for 4 bits and −128 to 127 for 8 bits.
The circle. Adding 1 to 0111 (7) gives 1000 (−8): the numbers “wrap around”. This is overflow.
With two's complement the computer adds negative numbers by ordinary binary addition. The carry out of the highest bit is discarded. Overflow happens when two numbers with the same sign give a result with the other sign.