If and summation: why do I have the index of the summation in the final result?

The other answer by Andrew technically answers your question, but, maybe what you really wanted was

Sum[If[b =!= 0, a[m], 0], {m, 1, 4}]

which returns

a[1] + a[2] + a[3] + a[4]  

The difference between the != and =!= operators is very important here. Because the truth value of b != 0 could not be determined, therefore the If[] expressions were returned unevaluated.

I suggest moving the If[] expression out of the summation. That is use

If[b =!= 0, Sum[a[m], {m, 1, 4}], 0]

instead. This simplifies the code and makes it easier to understand.

This works:

Sum[If[b != 0, a[m] // Evaluate, 0], {m, 1, 4}]

From the documentation:

If evaluates only the argument determined by the value of the condition.

$ $

You can use Evaluate to override HoldFirst etc. attributes of built-in functions.