>And the modem just transmits the digital signal to the appropriate destination(PC or router) and does no modulation? Am I right
Pretty much.
>How were the modems that were used(non DSL) earlier, different from these modems?
They took an analog signal and de-modulated it into a digital signal.
>They were to perform analog to digital conversion(as signal that travelled earlier were solely analog)?
Correct
>So is that due to this we were not able to make a call while connected to internet?
Correct