In the above working, which number is called the divisor? What is the value of 7/8 - 1/6?

In a division setup $A \div B$, $A$ is the dividend and $B$ is the divisor.
In the given long division problem:

      21
   _______
12 | 259
     -24
     ---
       19
      -12
      ---
        7

The number outside the division symbol, which is dividing 259, is the divisor.

The number 12 is dividing 259.
The divisor is $\boxed{\text{12}}$.

Multiples of 8: 8, 16, 24, 32, ...
Multiples of 6: 6, 12, 18, 24, 30, ...
The LCM of 8 and 6 is 24.

For $\frac{7}{8}$: Multiply the numerator and denominator by $24 \div 8 = 3$.

For $\frac{1}{6}$: Multiply the numerator and denominator by $24 \div 6 = 4$.

The value is $\boxed{\frac{17}{24}}$.