Here are some examples of eight-bit, twos complement binary addition. In each case, we compute the sum, and note if there was an overflow. If there was a carry out, the extra bit is shown on the next line. (It's falling into the bit bucket, where it will never be heard from again.) You can also look at the rules for determining overflow.
-39 + 92 = 53:
 1 1 1 1 1 1 0 1 1 0 0 1 + 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1
Carryout without overflow. Sum is correct.
-19 + -7 = -26:
 1 1 1 1 1 1 1 1 1 0 1 1 0 1 + 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0
Carryout without overflow. Sum is correct.
44 + 45 = 89:
 1 1 1 0 0 1 0 1 1 0 0 + 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 1
No overflow nor carryout.
104 + 45 = 149:
 1 1 1 0 1 1 0 1 0 0 0 + 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 1
Overflow, no carryout. Sum is not correct.
-75 + 59 = -16:
 1 1 1 1 1 1 1 0 1 1 0 1 0 1 + 0 0 1 1 1 0 1 1 1 1 1 1 0 0 0 0
No overflow nor carryout.
-103 + -69 = -172:
 1 1 1 1 1 1 1 0 0 1 1 0 0 1 + 1 0 1 1 1 0 1 1 0 1 0 1 0 1 0 0
Overflow, with incidental carryout. Sum is not correct.
10 + -3 = 7:
 1 1 1 1 1 0 0 0 0 1 0 1 0 + 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1
Carryout without overflow. Sum is correct.
127 + 1 = 128:
 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 + 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
Overflow, no carryout. Sum is not correct.
-1 + 1 = 0:
 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 + 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
Carryout without overflow. Sum is correct.