Two numbers r and s are relatively prime if there is no integer bigger than 1 that is both a divisor of r and a divisor of s. We also say in this case that r is relatively prime to s. For example, 18 and 8 are not relatively prime because 2 is a divisor of both of them. On the other hand, 9 and 8 are relatively prime because the only divisors common to both of them are 1 and -1. We never count 1 and -1 as common divisors when determining relative primality.
20 and 40 are not relatively prime--for example, 20 is a common divisor.
27 and 80 are relatively prime. The only divisors of 27 that are bigger than 1 are 3, 9, and 27, all powers of 3, and 3 doesn't divide 80.
17 is relatively prime to 27.
17 and 0 are not relatively prime because 17 is a divisor of both.
1 and 6 are relatively prime because the only positive divisor of 1 is 1.