Master Method to solve recurrences is 'a' related to 'b'?
No it's not always the case that $a=b$, since you might not necessarily use every sub-problem. Consider for example, the binary search algorithm. In the algorithm, you have a sorted array that you break into two sub-problems of the same size ($b=2$), but only recurse on one of them ($a=1$). In this case, the recurrence would look like: $$T(n) = T(n/2) + O(1)\,.$$

For an algorithm like merge-sort however, not only is the problem broken into two sub-problems of equal size, but you recurse on both sub-problems before merging the results. In this case, you have that $a = b = 2$, and so, the recurrence looks  like: $$T(n) = 2T(n/2) + O(n)\,.$$